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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    <a name="line.17"></a>
<FONT color="green">018</FONT>    package org.apache.commons.math.linear;<a name="line.18"></a>
<FONT color="green">019</FONT>    <a name="line.19"></a>
<FONT color="green">020</FONT>    import java.util.ArrayList;<a name="line.20"></a>
<FONT color="green">021</FONT>    import java.util.Arrays;<a name="line.21"></a>
<FONT color="green">022</FONT>    import java.util.List;<a name="line.22"></a>
<FONT color="green">023</FONT>    <a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math.ConvergenceException;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math.MathRuntimeException;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math.MaxIterationsExceededException;<a name="line.26"></a>
<FONT color="green">027</FONT>    import org.apache.commons.math.util.MathUtils;<a name="line.27"></a>
<FONT color="green">028</FONT>    <a name="line.28"></a>
<FONT color="green">029</FONT>    /**<a name="line.29"></a>
<FONT color="green">030</FONT>     * Calculates the eigen decomposition of a &lt;strong&gt;symmetric&lt;/strong&gt; matrix.<a name="line.30"></a>
<FONT color="green">031</FONT>     * &lt;p&gt;The eigen decomposition of matrix A is a set of two matrices:<a name="line.31"></a>
<FONT color="green">032</FONT>     * V and D such that A = V D V&lt;sup&gt;T&lt;/sup&gt;. A, V and D are all m &amp;times; m<a name="line.32"></a>
<FONT color="green">033</FONT>     * matrices.&lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * &lt;p&gt;As of 2.0, this class supports only &lt;strong&gt;symmetric&lt;/strong&gt; matrices,<a name="line.34"></a>
<FONT color="green">035</FONT>     * and hence computes only real realEigenvalues. This implies the D matrix returned by<a name="line.35"></a>
<FONT color="green">036</FONT>     * {@link #getD()} is always diagonal and the imaginary values returned {@link<a name="line.36"></a>
<FONT color="green">037</FONT>     * #getImagEigenvalue(int)} and {@link #getImagEigenvalues()} are always null.&lt;/p&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>     * &lt;p&gt;When called with a {@link RealMatrix} argument, this implementation only uses<a name="line.38"></a>
<FONT color="green">039</FONT>     * the upper part of the matrix, the part below the diagonal is not accessed at all.&lt;/p&gt;<a name="line.39"></a>
<FONT color="green">040</FONT>     * &lt;p&gt;Eigenvalues are computed as soon as the matrix is decomposed, but eigenvectors<a name="line.40"></a>
<FONT color="green">041</FONT>     * are computed only when required, i.e. only when one of the {@link #getEigenvector(int)},<a name="line.41"></a>
<FONT color="green">042</FONT>     * {@link #getV()}, {@link #getVT()}, {@link #getSolver()} methods is called.&lt;/p&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     * &lt;p&gt;This implementation is based on Inderjit Singh Dhillon thesis<a name="line.43"></a>
<FONT color="green">044</FONT>     * &lt;a href="http://www.cs.utexas.edu/users/inderjit/public_papers/thesis.pdf"&gt;A<a name="line.44"></a>
<FONT color="green">045</FONT>     * New O(n&lt;sup&gt;2&lt;/sup&gt;) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector<a name="line.45"></a>
<FONT color="green">046</FONT>     * Problem&lt;/a&gt;, on Beresford N. Parlett and Osni A. Marques paper &lt;a<a name="line.46"></a>
<FONT color="green">047</FONT>     * href="http://www.netlib.org/lapack/lawnspdf/lawn155.pdf"&gt;An Implementation of the<a name="line.47"></a>
<FONT color="green">048</FONT>     * dqds Algorithm (Positive Case)&lt;/a&gt; and on the corresponding LAPACK routines (DLARRE,<a name="line.48"></a>
<FONT color="green">049</FONT>     * DLASQ2, DLAZQ3, DLAZQ4, DLASQ5 and DLASQ6).&lt;/p&gt;<a name="line.49"></a>
<FONT color="green">050</FONT>     * @author Beresford Parlett, University of California, Berkeley, USA (fortran version)<a name="line.50"></a>
<FONT color="green">051</FONT>     * @author Jim Demmel, University of California, Berkeley, USA (fortran version)<a name="line.51"></a>
<FONT color="green">052</FONT>     * @author Inderjit Dhillon, University of Texas, Austin, USA(fortran version)<a name="line.52"></a>
<FONT color="green">053</FONT>     * @author Osni Marques, LBNL/NERSC, USA (fortran version)<a name="line.53"></a>
<FONT color="green">054</FONT>     * @author Christof Voemel, University of California, Berkeley, USA(fortran version)<a name="line.54"></a>
<FONT color="green">055</FONT>     * @version $Revision: 799857 $ $Date: 2009-08-01 09:07:12 -0400 (Sat, 01 Aug 2009) $<a name="line.55"></a>
<FONT color="green">056</FONT>     * @since 2.0<a name="line.56"></a>
<FONT color="green">057</FONT>     */<a name="line.57"></a>
<FONT color="green">058</FONT>    public class EigenDecompositionImpl implements EigenDecomposition {<a name="line.58"></a>
<FONT color="green">059</FONT>    <a name="line.59"></a>
<FONT color="green">060</FONT>        /** Tolerance. */<a name="line.60"></a>
<FONT color="green">061</FONT>        private static final double TOLERANCE = 100 * MathUtils.EPSILON;<a name="line.61"></a>
<FONT color="green">062</FONT>    <a name="line.62"></a>
<FONT color="green">063</FONT>        /** Squared tolerance. */<a name="line.63"></a>
<FONT color="green">064</FONT>        private static final double TOLERANCE_2 = TOLERANCE * TOLERANCE;<a name="line.64"></a>
<FONT color="green">065</FONT>    <a name="line.65"></a>
<FONT color="green">066</FONT>        /** Split tolerance. */<a name="line.66"></a>
<FONT color="green">067</FONT>        private double splitTolerance;<a name="line.67"></a>
<FONT color="green">068</FONT>    <a name="line.68"></a>
<FONT color="green">069</FONT>        /** Main diagonal of the tridiagonal matrix. */<a name="line.69"></a>
<FONT color="green">070</FONT>        private double[] main;<a name="line.70"></a>
<FONT color="green">071</FONT>    <a name="line.71"></a>
<FONT color="green">072</FONT>        /** Secondary diagonal of the tridiagonal matrix. */<a name="line.72"></a>
<FONT color="green">073</FONT>        private double[] secondary;<a name="line.73"></a>
<FONT color="green">074</FONT>    <a name="line.74"></a>
<FONT color="green">075</FONT>        /** Squared secondary diagonal of the tridiagonal matrix. */<a name="line.75"></a>
<FONT color="green">076</FONT>        private double[] squaredSecondary;<a name="line.76"></a>
<FONT color="green">077</FONT>    <a name="line.77"></a>
<FONT color="green">078</FONT>        /** Transformer to tridiagonal (may be null if matrix is already tridiagonal). */<a name="line.78"></a>
<FONT color="green">079</FONT>        private TriDiagonalTransformer transformer;<a name="line.79"></a>
<FONT color="green">080</FONT>    <a name="line.80"></a>
<FONT color="green">081</FONT>        /** Lower bound of spectra. */<a name="line.81"></a>
<FONT color="green">082</FONT>        private double lowerSpectra;<a name="line.82"></a>
<FONT color="green">083</FONT>    <a name="line.83"></a>
<FONT color="green">084</FONT>        /** Upper bound of spectra. */<a name="line.84"></a>
<FONT color="green">085</FONT>        private double upperSpectra;<a name="line.85"></a>
<FONT color="green">086</FONT>    <a name="line.86"></a>
<FONT color="green">087</FONT>        /** Minimum pivot in the Sturm sequence. */<a name="line.87"></a>
<FONT color="green">088</FONT>        private double minPivot;<a name="line.88"></a>
<FONT color="green">089</FONT>    <a name="line.89"></a>
<FONT color="green">090</FONT>        /** Current shift. */<a name="line.90"></a>
<FONT color="green">091</FONT>        private double sigma;<a name="line.91"></a>
<FONT color="green">092</FONT>    <a name="line.92"></a>
<FONT color="green">093</FONT>        /** Low part of the current shift. */<a name="line.93"></a>
<FONT color="green">094</FONT>        private double sigmaLow;<a name="line.94"></a>
<FONT color="green">095</FONT>    <a name="line.95"></a>
<FONT color="green">096</FONT>        /** Shift increment to apply. */<a name="line.96"></a>
<FONT color="green">097</FONT>        private double tau;<a name="line.97"></a>
<FONT color="green">098</FONT>    <a name="line.98"></a>
<FONT color="green">099</FONT>        /** Work array for all decomposition algorithms. */<a name="line.99"></a>
<FONT color="green">100</FONT>        private double[] work;<a name="line.100"></a>
<FONT color="green">101</FONT>    <a name="line.101"></a>
<FONT color="green">102</FONT>        /** Shift within qd array for ping-pong implementation. */<a name="line.102"></a>
<FONT color="green">103</FONT>        private int pingPong;<a name="line.103"></a>
<FONT color="green">104</FONT>    <a name="line.104"></a>
<FONT color="green">105</FONT>        /** Max value of diagonal elements in current segment. */<a name="line.105"></a>
<FONT color="green">106</FONT>        private double qMax;<a name="line.106"></a>
<FONT color="green">107</FONT>    <a name="line.107"></a>
<FONT color="green">108</FONT>        /** Min value of off-diagonal elements in current segment. */<a name="line.108"></a>
<FONT color="green">109</FONT>        private double eMin;<a name="line.109"></a>
<FONT color="green">110</FONT>    <a name="line.110"></a>
<FONT color="green">111</FONT>        /** Type of the last dqds shift. */<a name="line.111"></a>
<FONT color="green">112</FONT>        private int    tType;<a name="line.112"></a>
<FONT color="green">113</FONT>    <a name="line.113"></a>
<FONT color="green">114</FONT>        /** Minimal value on current state of the diagonal. */<a name="line.114"></a>
<FONT color="green">115</FONT>        private double dMin;<a name="line.115"></a>
<FONT color="green">116</FONT>    <a name="line.116"></a>
<FONT color="green">117</FONT>        /** Minimal value on current state of the diagonal, excluding last element. */<a name="line.117"></a>
<FONT color="green">118</FONT>        private double dMin1;<a name="line.118"></a>
<FONT color="green">119</FONT>    <a name="line.119"></a>
<FONT color="green">120</FONT>        /** Minimal value on current state of the diagonal, excluding last two elements. */<a name="line.120"></a>
<FONT color="green">121</FONT>        private double dMin2;<a name="line.121"></a>
<FONT color="green">122</FONT>    <a name="line.122"></a>
<FONT color="green">123</FONT>        /** Last value on current state of the diagonal. */<a name="line.123"></a>
<FONT color="green">124</FONT>        private double dN;<a name="line.124"></a>
<FONT color="green">125</FONT>    <a name="line.125"></a>
<FONT color="green">126</FONT>        /** Last but one value on current state of the diagonal. */<a name="line.126"></a>
<FONT color="green">127</FONT>        private double dN1;<a name="line.127"></a>
<FONT color="green">128</FONT>    <a name="line.128"></a>
<FONT color="green">129</FONT>        /** Last but two on current state of the diagonal. */<a name="line.129"></a>
<FONT color="green">130</FONT>        private double dN2;<a name="line.130"></a>
<FONT color="green">131</FONT>    <a name="line.131"></a>
<FONT color="green">132</FONT>        /** Shift ratio with respect to dMin used when tType == 6. */<a name="line.132"></a>
<FONT color="green">133</FONT>        private double g;<a name="line.133"></a>
<FONT color="green">134</FONT>    <a name="line.134"></a>
<FONT color="green">135</FONT>        /** Real part of the realEigenvalues. */<a name="line.135"></a>
<FONT color="green">136</FONT>        private double[] realEigenvalues;<a name="line.136"></a>
<FONT color="green">137</FONT>    <a name="line.137"></a>
<FONT color="green">138</FONT>        /** Imaginary part of the realEigenvalues. */<a name="line.138"></a>
<FONT color="green">139</FONT>        private double[] imagEigenvalues;<a name="line.139"></a>
<FONT color="green">140</FONT>    <a name="line.140"></a>
<FONT color="green">141</FONT>        /** Eigenvectors. */<a name="line.141"></a>
<FONT color="green">142</FONT>        private ArrayRealVector[] eigenvectors;<a name="line.142"></a>
<FONT color="green">143</FONT>    <a name="line.143"></a>
<FONT color="green">144</FONT>        /** Cached value of V. */<a name="line.144"></a>
<FONT color="green">145</FONT>        private RealMatrix cachedV;<a name="line.145"></a>
<FONT color="green">146</FONT>    <a name="line.146"></a>
<FONT color="green">147</FONT>        /** Cached value of D. */<a name="line.147"></a>
<FONT color="green">148</FONT>        private RealMatrix cachedD;<a name="line.148"></a>
<FONT color="green">149</FONT>    <a name="line.149"></a>
<FONT color="green">150</FONT>        /** Cached value of Vt. */<a name="line.150"></a>
<FONT color="green">151</FONT>        private RealMatrix cachedVt;<a name="line.151"></a>
<FONT color="green">152</FONT>    <a name="line.152"></a>
<FONT color="green">153</FONT>        /**<a name="line.153"></a>
<FONT color="green">154</FONT>         * Calculates the eigen decomposition of the given symmetric matrix. <a name="line.154"></a>
<FONT color="green">155</FONT>         * @param matrix The &lt;strong&gt;symmetric&lt;/strong&gt; matrix to decompose.<a name="line.155"></a>
<FONT color="green">156</FONT>         * @param splitTolerance tolerance on the off-diagonal elements relative to the<a name="line.156"></a>
<FONT color="green">157</FONT>         * geometric mean to split the tridiagonal matrix (a suggested value is<a name="line.157"></a>
<FONT color="green">158</FONT>         * {@link MathUtils#SAFE_MIN})<a name="line.158"></a>
<FONT color="green">159</FONT>         * @exception InvalidMatrixException (wrapping a {@link ConvergenceException}<a name="line.159"></a>
<FONT color="green">160</FONT>         * if algorithm fails to converge<a name="line.160"></a>
<FONT color="green">161</FONT>         */<a name="line.161"></a>
<FONT color="green">162</FONT>        public EigenDecompositionImpl(final RealMatrix matrix,<a name="line.162"></a>
<FONT color="green">163</FONT>                                      final double splitTolerance)<a name="line.163"></a>
<FONT color="green">164</FONT>            throws InvalidMatrixException {<a name="line.164"></a>
<FONT color="green">165</FONT>            if (isSymmetric(matrix)) {<a name="line.165"></a>
<FONT color="green">166</FONT>                this.splitTolerance = splitTolerance;<a name="line.166"></a>
<FONT color="green">167</FONT>                transformToTridiagonal(matrix);<a name="line.167"></a>
<FONT color="green">168</FONT>                decompose();<a name="line.168"></a>
<FONT color="green">169</FONT>            } else {<a name="line.169"></a>
<FONT color="green">170</FONT>                // as of 2.0, non-symmetric matrices (i.e. complex eigenvalues) are NOT supported<a name="line.170"></a>
<FONT color="green">171</FONT>                // see issue https://issues.apache.org/jira/browse/MATH-235<a name="line.171"></a>
<FONT color="green">172</FONT>                throw new InvalidMatrixException("eigen decomposition of assymetric matrices not supported yet");<a name="line.172"></a>
<FONT color="green">173</FONT>            }<a name="line.173"></a>
<FONT color="green">174</FONT>        }<a name="line.174"></a>
<FONT color="green">175</FONT>    <a name="line.175"></a>
<FONT color="green">176</FONT>        /**<a name="line.176"></a>
<FONT color="green">177</FONT>         * Calculates the eigen decomposition of the given tridiagonal symmetric matrix. <a name="line.177"></a>
<FONT color="green">178</FONT>         * @param main the main diagonal of the matrix (will be copied)<a name="line.178"></a>
<FONT color="green">179</FONT>         * @param secondary the secondary diagonal of the matrix (will be copied)<a name="line.179"></a>
<FONT color="green">180</FONT>         * @param splitTolerance tolerance on the off-diagonal elements relative to the<a name="line.180"></a>
<FONT color="green">181</FONT>         * geometric mean to split the tridiagonal matrix (a suggested value is<a name="line.181"></a>
<FONT color="green">182</FONT>         * {@link MathUtils#SAFE_MIN})<a name="line.182"></a>
<FONT color="green">183</FONT>         * @exception InvalidMatrixException (wrapping a {@link ConvergenceException}<a name="line.183"></a>
<FONT color="green">184</FONT>         * if algorithm fails to converge<a name="line.184"></a>
<FONT color="green">185</FONT>         */<a name="line.185"></a>
<FONT color="green">186</FONT>        public EigenDecompositionImpl(final double[] main, double[] secondary,<a name="line.186"></a>
<FONT color="green">187</FONT>                final double splitTolerance)<a name="line.187"></a>
<FONT color="green">188</FONT>            throws InvalidMatrixException {<a name="line.188"></a>
<FONT color="green">189</FONT>    <a name="line.189"></a>
<FONT color="green">190</FONT>            this.main      = main.clone();<a name="line.190"></a>
<FONT color="green">191</FONT>            this.secondary = secondary.clone();<a name="line.191"></a>
<FONT color="green">192</FONT>            transformer    = null;<a name="line.192"></a>
<FONT color="green">193</FONT>    <a name="line.193"></a>
<FONT color="green">194</FONT>            // pre-compute some elements<a name="line.194"></a>
<FONT color="green">195</FONT>            squaredSecondary = new double[secondary.length];<a name="line.195"></a>
<FONT color="green">196</FONT>            for (int i = 0; i &lt; squaredSecondary.length; ++i) {<a name="line.196"></a>
<FONT color="green">197</FONT>                final double s = secondary[i];<a name="line.197"></a>
<FONT color="green">198</FONT>                squaredSecondary[i] = s * s;<a name="line.198"></a>
<FONT color="green">199</FONT>            }<a name="line.199"></a>
<FONT color="green">200</FONT>    <a name="line.200"></a>
<FONT color="green">201</FONT>            this.splitTolerance = splitTolerance;<a name="line.201"></a>
<FONT color="green">202</FONT>            decompose();<a name="line.202"></a>
<FONT color="green">203</FONT>    <a name="line.203"></a>
<FONT color="green">204</FONT>        }<a name="line.204"></a>
<FONT color="green">205</FONT>    <a name="line.205"></a>
<FONT color="green">206</FONT>        /**<a name="line.206"></a>
<FONT color="green">207</FONT>         * Check if a matrix is symmetric.<a name="line.207"></a>
<FONT color="green">208</FONT>         * @param matrix matrix to check<a name="line.208"></a>
<FONT color="green">209</FONT>         * @return true if matrix is symmetric<a name="line.209"></a>
<FONT color="green">210</FONT>         */<a name="line.210"></a>
<FONT color="green">211</FONT>        private boolean isSymmetric(final RealMatrix matrix) {<a name="line.211"></a>
<FONT color="green">212</FONT>            final int rows    = matrix.getRowDimension();<a name="line.212"></a>
<FONT color="green">213</FONT>            final int columns = matrix.getColumnDimension();<a name="line.213"></a>
<FONT color="green">214</FONT>            final double eps  = 10 * rows * columns * MathUtils.EPSILON;<a name="line.214"></a>
<FONT color="green">215</FONT>            for (int i = 0; i &lt; rows; ++i) {<a name="line.215"></a>
<FONT color="green">216</FONT>                for (int j = i + 1; j &lt; columns; ++j) {<a name="line.216"></a>
<FONT color="green">217</FONT>                    final double mij = matrix.getEntry(i, j);<a name="line.217"></a>
<FONT color="green">218</FONT>                    final double mji = matrix.getEntry(j, i);<a name="line.218"></a>
<FONT color="green">219</FONT>                    if (Math.abs(mij - mji) &gt; (Math.max(Math.abs(mij), Math.abs(mji)) * eps)) {<a name="line.219"></a>
<FONT color="green">220</FONT>                        return false;<a name="line.220"></a>
<FONT color="green">221</FONT>                    }<a name="line.221"></a>
<FONT color="green">222</FONT>                }<a name="line.222"></a>
<FONT color="green">223</FONT>            }<a name="line.223"></a>
<FONT color="green">224</FONT>            return true;<a name="line.224"></a>
<FONT color="green">225</FONT>        }<a name="line.225"></a>
<FONT color="green">226</FONT>    <a name="line.226"></a>
<FONT color="green">227</FONT>        /**<a name="line.227"></a>
<FONT color="green">228</FONT>         * Decompose a tridiagonal symmetric matrix. <a name="line.228"></a>
<FONT color="green">229</FONT>         * @exception InvalidMatrixException (wrapping a {@link ConvergenceException}<a name="line.229"></a>
<FONT color="green">230</FONT>         * if algorithm fails to converge<a name="line.230"></a>
<FONT color="green">231</FONT>         */<a name="line.231"></a>
<FONT color="green">232</FONT>        private void decompose() {<a name="line.232"></a>
<FONT color="green">233</FONT>    <a name="line.233"></a>
<FONT color="green">234</FONT>            cachedV  = null;<a name="line.234"></a>
<FONT color="green">235</FONT>            cachedD  = null;<a name="line.235"></a>
<FONT color="green">236</FONT>            cachedVt = null;<a name="line.236"></a>
<FONT color="green">237</FONT>            work     = new double[6 * main.length];<a name="line.237"></a>
<FONT color="green">238</FONT>    <a name="line.238"></a>
<FONT color="green">239</FONT>            // compute the Gershgorin circles<a name="line.239"></a>
<FONT color="green">240</FONT>            computeGershgorinCircles();<a name="line.240"></a>
<FONT color="green">241</FONT>    <a name="line.241"></a>
<FONT color="green">242</FONT>            // find all the realEigenvalues<a name="line.242"></a>
<FONT color="green">243</FONT>            findEigenvalues();<a name="line.243"></a>
<FONT color="green">244</FONT>    <a name="line.244"></a>
<FONT color="green">245</FONT>            // we will search for eigenvectors only if required<a name="line.245"></a>
<FONT color="green">246</FONT>            eigenvectors = null;<a name="line.246"></a>
<FONT color="green">247</FONT>    <a name="line.247"></a>
<FONT color="green">248</FONT>        }<a name="line.248"></a>
<FONT color="green">249</FONT>    <a name="line.249"></a>
<FONT color="green">250</FONT>        /** {@inheritDoc} */<a name="line.250"></a>
<FONT color="green">251</FONT>        public RealMatrix getV()<a name="line.251"></a>
<FONT color="green">252</FONT>            throws InvalidMatrixException {<a name="line.252"></a>
<FONT color="green">253</FONT>    <a name="line.253"></a>
<FONT color="green">254</FONT>            if (cachedV == null) {<a name="line.254"></a>
<FONT color="green">255</FONT>    <a name="line.255"></a>
<FONT color="green">256</FONT>                if (eigenvectors == null) {<a name="line.256"></a>
<FONT color="green">257</FONT>                    findEigenVectors();<a name="line.257"></a>
<FONT color="green">258</FONT>                }<a name="line.258"></a>
<FONT color="green">259</FONT>    <a name="line.259"></a>
<FONT color="green">260</FONT>                final int m = eigenvectors.length;<a name="line.260"></a>
<FONT color="green">261</FONT>                cachedV = MatrixUtils.createRealMatrix(m, m);<a name="line.261"></a>
<FONT color="green">262</FONT>                for (int k = 0; k &lt; m; ++k) {<a name="line.262"></a>
<FONT color="green">263</FONT>                    cachedV.setColumnVector(k, eigenvectors[k]);<a name="line.263"></a>
<FONT color="green">264</FONT>                }<a name="line.264"></a>
<FONT color="green">265</FONT>    <a name="line.265"></a>
<FONT color="green">266</FONT>            }<a name="line.266"></a>
<FONT color="green">267</FONT>    <a name="line.267"></a>
<FONT color="green">268</FONT>            // return the cached matrix<a name="line.268"></a>
<FONT color="green">269</FONT>            return cachedV;<a name="line.269"></a>
<FONT color="green">270</FONT>    <a name="line.270"></a>
<FONT color="green">271</FONT>        }<a name="line.271"></a>
<FONT color="green">272</FONT>    <a name="line.272"></a>
<FONT color="green">273</FONT>        /** {@inheritDoc} */<a name="line.273"></a>
<FONT color="green">274</FONT>        public RealMatrix getD()<a name="line.274"></a>
<FONT color="green">275</FONT>            throws InvalidMatrixException {<a name="line.275"></a>
<FONT color="green">276</FONT>            if (cachedD == null) {<a name="line.276"></a>
<FONT color="green">277</FONT>                // cache the matrix for subsequent calls<a name="line.277"></a>
<FONT color="green">278</FONT>                cachedD = MatrixUtils.createRealDiagonalMatrix(realEigenvalues);<a name="line.278"></a>
<FONT color="green">279</FONT>            }<a name="line.279"></a>
<FONT color="green">280</FONT>            return cachedD;<a name="line.280"></a>
<FONT color="green">281</FONT>        }<a name="line.281"></a>
<FONT color="green">282</FONT>    <a name="line.282"></a>
<FONT color="green">283</FONT>        /** {@inheritDoc} */<a name="line.283"></a>
<FONT color="green">284</FONT>        public RealMatrix getVT()<a name="line.284"></a>
<FONT color="green">285</FONT>            throws InvalidMatrixException {<a name="line.285"></a>
<FONT color="green">286</FONT>    <a name="line.286"></a>
<FONT color="green">287</FONT>            if (cachedVt == null) {<a name="line.287"></a>
<FONT color="green">288</FONT>    <a name="line.288"></a>
<FONT color="green">289</FONT>                if (eigenvectors == null) {<a name="line.289"></a>
<FONT color="green">290</FONT>                    findEigenVectors();<a name="line.290"></a>
<FONT color="green">291</FONT>                }<a name="line.291"></a>
<FONT color="green">292</FONT>    <a name="line.292"></a>
<FONT color="green">293</FONT>                final int m = eigenvectors.length;<a name="line.293"></a>
<FONT color="green">294</FONT>                cachedVt = MatrixUtils.createRealMatrix(m, m);<a name="line.294"></a>
<FONT color="green">295</FONT>                for (int k = 0; k &lt; m; ++k) {<a name="line.295"></a>
<FONT color="green">296</FONT>                    cachedVt.setRowVector(k, eigenvectors[k]);<a name="line.296"></a>
<FONT color="green">297</FONT>                }<a name="line.297"></a>
<FONT color="green">298</FONT>    <a name="line.298"></a>
<FONT color="green">299</FONT>            }<a name="line.299"></a>
<FONT color="green">300</FONT>    <a name="line.300"></a>
<FONT color="green">301</FONT>            // return the cached matrix<a name="line.301"></a>
<FONT color="green">302</FONT>            return cachedVt;<a name="line.302"></a>
<FONT color="green">303</FONT>    <a name="line.303"></a>
<FONT color="green">304</FONT>        }<a name="line.304"></a>
<FONT color="green">305</FONT>    <a name="line.305"></a>
<FONT color="green">306</FONT>        /** {@inheritDoc} */<a name="line.306"></a>
<FONT color="green">307</FONT>        public double[] getRealEigenvalues()<a name="line.307"></a>
<FONT color="green">308</FONT>            throws InvalidMatrixException {<a name="line.308"></a>
<FONT color="green">309</FONT>            return realEigenvalues.clone();<a name="line.309"></a>
<FONT color="green">310</FONT>        }<a name="line.310"></a>
<FONT color="green">311</FONT>    <a name="line.311"></a>
<FONT color="green">312</FONT>        /** {@inheritDoc} */<a name="line.312"></a>
<FONT color="green">313</FONT>        public double getRealEigenvalue(final int i)<a name="line.313"></a>
<FONT color="green">314</FONT>            throws InvalidMatrixException, ArrayIndexOutOfBoundsException {<a name="line.314"></a>
<FONT color="green">315</FONT>            return realEigenvalues[i];<a name="line.315"></a>
<FONT color="green">316</FONT>        }<a name="line.316"></a>
<FONT color="green">317</FONT>    <a name="line.317"></a>
<FONT color="green">318</FONT>        /** {@inheritDoc} */<a name="line.318"></a>
<FONT color="green">319</FONT>        public double[] getImagEigenvalues()<a name="line.319"></a>
<FONT color="green">320</FONT>            throws InvalidMatrixException {<a name="line.320"></a>
<FONT color="green">321</FONT>            return imagEigenvalues.clone();<a name="line.321"></a>
<FONT color="green">322</FONT>        }<a name="line.322"></a>
<FONT color="green">323</FONT>    <a name="line.323"></a>
<FONT color="green">324</FONT>        /** {@inheritDoc} */<a name="line.324"></a>
<FONT color="green">325</FONT>        public double getImagEigenvalue(final int i)<a name="line.325"></a>
<FONT color="green">326</FONT>            throws InvalidMatrixException, ArrayIndexOutOfBoundsException {<a name="line.326"></a>
<FONT color="green">327</FONT>            return imagEigenvalues[i];<a name="line.327"></a>
<FONT color="green">328</FONT>        }<a name="line.328"></a>
<FONT color="green">329</FONT>    <a name="line.329"></a>
<FONT color="green">330</FONT>        /** {@inheritDoc} */<a name="line.330"></a>
<FONT color="green">331</FONT>        public RealVector getEigenvector(final int i)<a name="line.331"></a>
<FONT color="green">332</FONT>            throws InvalidMatrixException, ArrayIndexOutOfBoundsException {<a name="line.332"></a>
<FONT color="green">333</FONT>            if (eigenvectors == null) {<a name="line.333"></a>
<FONT color="green">334</FONT>                findEigenVectors();<a name="line.334"></a>
<FONT color="green">335</FONT>            }<a name="line.335"></a>
<FONT color="green">336</FONT>            return eigenvectors[i].copy();<a name="line.336"></a>
<FONT color="green">337</FONT>        }<a name="line.337"></a>
<FONT color="green">338</FONT>    <a name="line.338"></a>
<FONT color="green">339</FONT>        /**<a name="line.339"></a>
<FONT color="green">340</FONT>         * Return the determinant of the matrix<a name="line.340"></a>
<FONT color="green">341</FONT>         * @return determinant of the matrix<a name="line.341"></a>
<FONT color="green">342</FONT>         */<a name="line.342"></a>
<FONT color="green">343</FONT>        public double getDeterminant() {<a name="line.343"></a>
<FONT color="green">344</FONT>            double determinant = 1;<a name="line.344"></a>
<FONT color="green">345</FONT>            for (double lambda : realEigenvalues) {<a name="line.345"></a>
<FONT color="green">346</FONT>                determinant *= lambda;<a name="line.346"></a>
<FONT color="green">347</FONT>            }<a name="line.347"></a>
<FONT color="green">348</FONT>            return determinant;<a name="line.348"></a>
<FONT color="green">349</FONT>        }<a name="line.349"></a>
<FONT color="green">350</FONT>    <a name="line.350"></a>
<FONT color="green">351</FONT>        /** {@inheritDoc} */<a name="line.351"></a>
<FONT color="green">352</FONT>        public DecompositionSolver getSolver() {<a name="line.352"></a>
<FONT color="green">353</FONT>            if (eigenvectors == null) {<a name="line.353"></a>
<FONT color="green">354</FONT>                findEigenVectors();<a name="line.354"></a>
<FONT color="green">355</FONT>            }<a name="line.355"></a>
<FONT color="green">356</FONT>            return new Solver(realEigenvalues, imagEigenvalues, eigenvectors);<a name="line.356"></a>
<FONT color="green">357</FONT>        }<a name="line.357"></a>
<FONT color="green">358</FONT>    <a name="line.358"></a>
<FONT color="green">359</FONT>        /** Specialized solver. */<a name="line.359"></a>
<FONT color="green">360</FONT>        private static class Solver implements DecompositionSolver {<a name="line.360"></a>
<FONT color="green">361</FONT>        <a name="line.361"></a>
<FONT color="green">362</FONT>            /** Real part of the realEigenvalues. */<a name="line.362"></a>
<FONT color="green">363</FONT>            private double[] realEigenvalues;<a name="line.363"></a>
<FONT color="green">364</FONT>    <a name="line.364"></a>
<FONT color="green">365</FONT>            /** Imaginary part of the realEigenvalues. */<a name="line.365"></a>
<FONT color="green">366</FONT>            private double[] imagEigenvalues;<a name="line.366"></a>
<FONT color="green">367</FONT>    <a name="line.367"></a>
<FONT color="green">368</FONT>            /** Eigenvectors. */<a name="line.368"></a>
<FONT color="green">369</FONT>            private final ArrayRealVector[] eigenvectors;<a name="line.369"></a>
<FONT color="green">370</FONT>    <a name="line.370"></a>
<FONT color="green">371</FONT>            /**<a name="line.371"></a>
<FONT color="green">372</FONT>             * Build a solver from decomposed matrix.<a name="line.372"></a>
<FONT color="green">373</FONT>             * @param realEigenvalues real parts of the eigenvalues<a name="line.373"></a>
<FONT color="green">374</FONT>             * @param imagEigenvalues imaginary parts of the eigenvalues<a name="line.374"></a>
<FONT color="green">375</FONT>             * @param eigenvectors eigenvectors<a name="line.375"></a>
<FONT color="green">376</FONT>             */<a name="line.376"></a>
<FONT color="green">377</FONT>            private Solver(final double[] realEigenvalues, final double[] imagEigenvalues,<a name="line.377"></a>
<FONT color="green">378</FONT>                           final ArrayRealVector[] eigenvectors) {<a name="line.378"></a>
<FONT color="green">379</FONT>                this.realEigenvalues = realEigenvalues;<a name="line.379"></a>
<FONT color="green">380</FONT>                this.imagEigenvalues = imagEigenvalues;<a name="line.380"></a>
<FONT color="green">381</FONT>                this.eigenvectors    = eigenvectors; <a name="line.381"></a>
<FONT color="green">382</FONT>            }<a name="line.382"></a>
<FONT color="green">383</FONT>    <a name="line.383"></a>
<FONT color="green">384</FONT>            /** Solve the linear equation A &amp;times; X = B for symmetric matrices A.<a name="line.384"></a>
<FONT color="green">385</FONT>             * &lt;p&gt;This method only find exact linear solutions, i.e. solutions for<a name="line.385"></a>
<FONT color="green">386</FONT>             * which ||A &amp;times; X - B|| is exactly 0.&lt;/p&gt;<a name="line.386"></a>
<FONT color="green">387</FONT>             * @param b right-hand side of the equation A &amp;times; X = B<a name="line.387"></a>
<FONT color="green">388</FONT>             * @return a vector X that minimizes the two norm of A &amp;times; X - B<a name="line.388"></a>
<FONT color="green">389</FONT>             * @exception IllegalArgumentException if matrices dimensions don't match<a name="line.389"></a>
<FONT color="green">390</FONT>             * @exception InvalidMatrixException if decomposed matrix is singular<a name="line.390"></a>
<FONT color="green">391</FONT>             */<a name="line.391"></a>
<FONT color="green">392</FONT>            public double[] solve(final double[] b)<a name="line.392"></a>
<FONT color="green">393</FONT>                throws IllegalArgumentException, InvalidMatrixException {<a name="line.393"></a>
<FONT color="green">394</FONT>    <a name="line.394"></a>
<FONT color="green">395</FONT>                if (!isNonSingular()) {<a name="line.395"></a>
<FONT color="green">396</FONT>                    throw new SingularMatrixException();<a name="line.396"></a>
<FONT color="green">397</FONT>                }<a name="line.397"></a>
<FONT color="green">398</FONT>    <a name="line.398"></a>
<FONT color="green">399</FONT>                final int m = realEigenvalues.length;<a name="line.399"></a>
<FONT color="green">400</FONT>                if (b.length != m) {<a name="line.400"></a>
<FONT color="green">401</FONT>                    throw MathRuntimeException.createIllegalArgumentException(<a name="line.401"></a>
<FONT color="green">402</FONT>                            "vector length mismatch: got {0} but expected {1}",<a name="line.402"></a>
<FONT color="green">403</FONT>                            b.length, m);<a name="line.403"></a>
<FONT color="green">404</FONT>                }<a name="line.404"></a>
<FONT color="green">405</FONT>    <a name="line.405"></a>
<FONT color="green">406</FONT>                final double[] bp = new double[m];<a name="line.406"></a>
<FONT color="green">407</FONT>                for (int i = 0; i &lt; m; ++i) {<a name="line.407"></a>
<FONT color="green">408</FONT>                    final ArrayRealVector v = eigenvectors[i];<a name="line.408"></a>
<FONT color="green">409</FONT>                    final double[] vData = v.getDataRef();<a name="line.409"></a>
<FONT color="green">410</FONT>                    final double s = v.dotProduct(b) / realEigenvalues[i];<a name="line.410"></a>
<FONT color="green">411</FONT>                    for (int j = 0; j &lt; m; ++j) {<a name="line.411"></a>
<FONT color="green">412</FONT>                        bp[j] += s * vData[j];<a name="line.412"></a>
<FONT color="green">413</FONT>                    }<a name="line.413"></a>
<FONT color="green">414</FONT>                }<a name="line.414"></a>
<FONT color="green">415</FONT>    <a name="line.415"></a>
<FONT color="green">416</FONT>                return bp;<a name="line.416"></a>
<FONT color="green">417</FONT>    <a name="line.417"></a>
<FONT color="green">418</FONT>            }<a name="line.418"></a>
<FONT color="green">419</FONT>    <a name="line.419"></a>
<FONT color="green">420</FONT>            /** Solve the linear equation A &amp;times; X = B for symmetric matrices A.<a name="line.420"></a>
<FONT color="green">421</FONT>             * &lt;p&gt;This method only find exact linear solutions, i.e. solutions for<a name="line.421"></a>
<FONT color="green">422</FONT>             * which ||A &amp;times; X - B|| is exactly 0.&lt;/p&gt;<a name="line.422"></a>
<FONT color="green">423</FONT>             * @param b right-hand side of the equation A &amp;times; X = B<a name="line.423"></a>
<FONT color="green">424</FONT>             * @return a vector X that minimizes the two norm of A &amp;times; X - B<a name="line.424"></a>
<FONT color="green">425</FONT>             * @exception IllegalArgumentException if matrices dimensions don't match<a name="line.425"></a>
<FONT color="green">426</FONT>             * @exception InvalidMatrixException if decomposed matrix is singular<a name="line.426"></a>
<FONT color="green">427</FONT>             */<a name="line.427"></a>
<FONT color="green">428</FONT>            public RealVector solve(final RealVector b)<a name="line.428"></a>
<FONT color="green">429</FONT>                throws IllegalArgumentException, InvalidMatrixException {<a name="line.429"></a>
<FONT color="green">430</FONT>    <a name="line.430"></a>
<FONT color="green">431</FONT>                if (!isNonSingular()) {<a name="line.431"></a>
<FONT color="green">432</FONT>                    throw new SingularMatrixException();<a name="line.432"></a>
<FONT color="green">433</FONT>                }<a name="line.433"></a>
<FONT color="green">434</FONT>    <a name="line.434"></a>
<FONT color="green">435</FONT>                final int m = realEigenvalues.length;<a name="line.435"></a>
<FONT color="green">436</FONT>                if (b.getDimension() != m) {<a name="line.436"></a>
<FONT color="green">437</FONT>                    throw MathRuntimeException.createIllegalArgumentException(<a name="line.437"></a>
<FONT color="green">438</FONT>                            "vector length mismatch: got {0} but expected {1}",<a name="line.438"></a>
<FONT color="green">439</FONT>                            b.getDimension(), m);<a name="line.439"></a>
<FONT color="green">440</FONT>                }<a name="line.440"></a>
<FONT color="green">441</FONT>    <a name="line.441"></a>
<FONT color="green">442</FONT>                final double[] bp = new double[m];<a name="line.442"></a>
<FONT color="green">443</FONT>                for (int i = 0; i &lt; m; ++i) {<a name="line.443"></a>
<FONT color="green">444</FONT>                    final ArrayRealVector v = eigenvectors[i];<a name="line.444"></a>
<FONT color="green">445</FONT>                    final double[] vData = v.getDataRef();<a name="line.445"></a>
<FONT color="green">446</FONT>                    final double s = v.dotProduct(b) / realEigenvalues[i];<a name="line.446"></a>
<FONT color="green">447</FONT>                    for (int j = 0; j &lt; m; ++j) {<a name="line.447"></a>
<FONT color="green">448</FONT>                        bp[j] += s * vData[j];<a name="line.448"></a>
<FONT color="green">449</FONT>                    }<a name="line.449"></a>
<FONT color="green">450</FONT>                }<a name="line.450"></a>
<FONT color="green">451</FONT>    <a name="line.451"></a>
<FONT color="green">452</FONT>                return new ArrayRealVector(bp, false);<a name="line.452"></a>
<FONT color="green">453</FONT>    <a name="line.453"></a>
<FONT color="green">454</FONT>            }<a name="line.454"></a>
<FONT color="green">455</FONT>    <a name="line.455"></a>
<FONT color="green">456</FONT>            /** Solve the linear equation A &amp;times; X = B for symmetric matrices A.<a name="line.456"></a>
<FONT color="green">457</FONT>             * &lt;p&gt;This method only find exact linear solutions, i.e. solutions for<a name="line.457"></a>
<FONT color="green">458</FONT>             * which ||A &amp;times; X - B|| is exactly 0.&lt;/p&gt;<a name="line.458"></a>
<FONT color="green">459</FONT>             * @param b right-hand side of the equation A &amp;times; X = B<a name="line.459"></a>
<FONT color="green">460</FONT>             * @return a matrix X that minimizes the two norm of A &amp;times; X - B<a name="line.460"></a>
<FONT color="green">461</FONT>             * @exception IllegalArgumentException if matrices dimensions don't match<a name="line.461"></a>
<FONT color="green">462</FONT>             * @exception InvalidMatrixException if decomposed matrix is singular<a name="line.462"></a>
<FONT color="green">463</FONT>             */<a name="line.463"></a>
<FONT color="green">464</FONT>            public RealMatrix solve(final RealMatrix b)<a name="line.464"></a>
<FONT color="green">465</FONT>                throws IllegalArgumentException, InvalidMatrixException {<a name="line.465"></a>
<FONT color="green">466</FONT>    <a name="line.466"></a>
<FONT color="green">467</FONT>                if (!isNonSingular()) {<a name="line.467"></a>
<FONT color="green">468</FONT>                    throw new SingularMatrixException();<a name="line.468"></a>
<FONT color="green">469</FONT>                }<a name="line.469"></a>
<FONT color="green">470</FONT>    <a name="line.470"></a>
<FONT color="green">471</FONT>                final int m = realEigenvalues.length;<a name="line.471"></a>
<FONT color="green">472</FONT>                if (b.getRowDimension() != m) {<a name="line.472"></a>
<FONT color="green">473</FONT>                    throw MathRuntimeException.createIllegalArgumentException(<a name="line.473"></a>
<FONT color="green">474</FONT>                            "dimensions mismatch: got {0}x{1} but expected {2}x{3}",<a name="line.474"></a>
<FONT color="green">475</FONT>                            b.getRowDimension(), b.getColumnDimension(), m, "n");<a name="line.475"></a>
<FONT color="green">476</FONT>                }<a name="line.476"></a>
<FONT color="green">477</FONT>    <a name="line.477"></a>
<FONT color="green">478</FONT>                final int nColB = b.getColumnDimension();<a name="line.478"></a>
<FONT color="green">479</FONT>                final double[][] bp = new double[m][nColB];<a name="line.479"></a>
<FONT color="green">480</FONT>                for (int k = 0; k &lt; nColB; ++k) {<a name="line.480"></a>
<FONT color="green">481</FONT>                    for (int i = 0; i &lt; m; ++i) {<a name="line.481"></a>
<FONT color="green">482</FONT>                        final ArrayRealVector v = eigenvectors[i];<a name="line.482"></a>
<FONT color="green">483</FONT>                        final double[] vData = v.getDataRef();<a name="line.483"></a>
<FONT color="green">484</FONT>                        double s = 0;<a name="line.484"></a>
<FONT color="green">485</FONT>                        for (int j = 0; j &lt; m; ++j) {<a name="line.485"></a>
<FONT color="green">486</FONT>                            s += v.getEntry(j) * b.getEntry(j, k);<a name="line.486"></a>
<FONT color="green">487</FONT>                        }<a name="line.487"></a>
<FONT color="green">488</FONT>                        s /= realEigenvalues[i];<a name="line.488"></a>
<FONT color="green">489</FONT>                        for (int j = 0; j &lt; m; ++j) {<a name="line.489"></a>
<FONT color="green">490</FONT>                            bp[j][k] += s * vData[j];<a name="line.490"></a>
<FONT color="green">491</FONT>                        }<a name="line.491"></a>
<FONT color="green">492</FONT>                    }<a name="line.492"></a>
<FONT color="green">493</FONT>                }<a name="line.493"></a>
<FONT color="green">494</FONT>    <a name="line.494"></a>
<FONT color="green">495</FONT>                return MatrixUtils.createRealMatrix(bp);<a name="line.495"></a>
<FONT color="green">496</FONT>    <a name="line.496"></a>
<FONT color="green">497</FONT>            }<a name="line.497"></a>
<FONT color="green">498</FONT>    <a name="line.498"></a>
<FONT color="green">499</FONT>            /**<a name="line.499"></a>
<FONT color="green">500</FONT>             * Check if the decomposed matrix is non-singular.<a name="line.500"></a>
<FONT color="green">501</FONT>             * @return true if the decomposed matrix is non-singular<a name="line.501"></a>
<FONT color="green">502</FONT>             */<a name="line.502"></a>
<FONT color="green">503</FONT>            public boolean isNonSingular() {<a name="line.503"></a>
<FONT color="green">504</FONT>                for (int i = 0; i &lt; realEigenvalues.length; ++i) {<a name="line.504"></a>
<FONT color="green">505</FONT>                    if ((realEigenvalues[i] == 0) &amp;&amp; (imagEigenvalues[i] == 0)) {<a name="line.505"></a>
<FONT color="green">506</FONT>                        return false;<a name="line.506"></a>
<FONT color="green">507</FONT>                    }<a name="line.507"></a>
<FONT color="green">508</FONT>                }<a name="line.508"></a>
<FONT color="green">509</FONT>                return true;<a name="line.509"></a>
<FONT color="green">510</FONT>            }<a name="line.510"></a>
<FONT color="green">511</FONT>    <a name="line.511"></a>
<FONT color="green">512</FONT>            /** Get the inverse of the decomposed matrix.<a name="line.512"></a>
<FONT color="green">513</FONT>             * @return inverse matrix<a name="line.513"></a>
<FONT color="green">514</FONT>             * @throws InvalidMatrixException if decomposed matrix is singular<a name="line.514"></a>
<FONT color="green">515</FONT>             */<a name="line.515"></a>
<FONT color="green">516</FONT>            public RealMatrix getInverse()<a name="line.516"></a>
<FONT color="green">517</FONT>                throws InvalidMatrixException {<a name="line.517"></a>
<FONT color="green">518</FONT>    <a name="line.518"></a>
<FONT color="green">519</FONT>                if (!isNonSingular()) {<a name="line.519"></a>
<FONT color="green">520</FONT>                    throw new SingularMatrixException();<a name="line.520"></a>
<FONT color="green">521</FONT>                }<a name="line.521"></a>
<FONT color="green">522</FONT>    <a name="line.522"></a>
<FONT color="green">523</FONT>                final int m = realEigenvalues.length;<a name="line.523"></a>
<FONT color="green">524</FONT>                final double[][] invData = new double[m][m];<a name="line.524"></a>
<FONT color="green">525</FONT>    <a name="line.525"></a>
<FONT color="green">526</FONT>                for (int i = 0; i &lt; m; ++i) {<a name="line.526"></a>
<FONT color="green">527</FONT>                    final double[] invI = invData[i];<a name="line.527"></a>
<FONT color="green">528</FONT>                    for (int j = 0; j &lt; m; ++j) {<a name="line.528"></a>
<FONT color="green">529</FONT>                        double invIJ = 0;<a name="line.529"></a>
<FONT color="green">530</FONT>                        for (int k = 0; k &lt; m; ++k) {<a name="line.530"></a>
<FONT color="green">531</FONT>                            final double[] vK = eigenvectors[k].getDataRef();<a name="line.531"></a>
<FONT color="green">532</FONT>                            invIJ += vK[i] * vK[j] / realEigenvalues[k];<a name="line.532"></a>
<FONT color="green">533</FONT>                        }<a name="line.533"></a>
<FONT color="green">534</FONT>                        invI[j] = invIJ;<a name="line.534"></a>
<FONT color="green">535</FONT>                    }<a name="line.535"></a>
<FONT color="green">536</FONT>                }<a name="line.536"></a>
<FONT color="green">537</FONT>                return MatrixUtils.createRealMatrix(invData);<a name="line.537"></a>
<FONT color="green">538</FONT>    <a name="line.538"></a>
<FONT color="green">539</FONT>            }<a name="line.539"></a>
<FONT color="green">540</FONT>    <a name="line.540"></a>
<FONT color="green">541</FONT>        }<a name="line.541"></a>
<FONT color="green">542</FONT>    <a name="line.542"></a>
<FONT color="green">543</FONT>        /**<a name="line.543"></a>
<FONT color="green">544</FONT>         * Transform matrix to tridiagonal.<a name="line.544"></a>
<FONT color="green">545</FONT>         * @param matrix matrix to transform<a name="line.545"></a>
<FONT color="green">546</FONT>         */<a name="line.546"></a>
<FONT color="green">547</FONT>        private void transformToTridiagonal(final RealMatrix matrix) {<a name="line.547"></a>
<FONT color="green">548</FONT>    <a name="line.548"></a>
<FONT color="green">549</FONT>            // transform the matrix to tridiagonal<a name="line.549"></a>
<FONT color="green">550</FONT>            transformer = new TriDiagonalTransformer(matrix);<a name="line.550"></a>
<FONT color="green">551</FONT>            main      = transformer.getMainDiagonalRef();<a name="line.551"></a>
<FONT color="green">552</FONT>            secondary = transformer.getSecondaryDiagonalRef();<a name="line.552"></a>
<FONT color="green">553</FONT>    <a name="line.553"></a>
<FONT color="green">554</FONT>            // pre-compute some elements<a name="line.554"></a>
<FONT color="green">555</FONT>            squaredSecondary = new double[secondary.length];<a name="line.555"></a>
<FONT color="green">556</FONT>            for (int i = 0; i &lt; squaredSecondary.length; ++i) {<a name="line.556"></a>
<FONT color="green">557</FONT>                final double s = secondary[i];<a name="line.557"></a>
<FONT color="green">558</FONT>                squaredSecondary[i] = s * s;<a name="line.558"></a>
<FONT color="green">559</FONT>            }<a name="line.559"></a>
<FONT color="green">560</FONT>    <a name="line.560"></a>
<FONT color="green">561</FONT>        }<a name="line.561"></a>
<FONT color="green">562</FONT>    <a name="line.562"></a>
<FONT color="green">563</FONT>        /**<a name="line.563"></a>
<FONT color="green">564</FONT>         * Compute the Gershgorin circles for all rows.<a name="line.564"></a>
<FONT color="green">565</FONT>         */<a name="line.565"></a>
<FONT color="green">566</FONT>        private void computeGershgorinCircles() {<a name="line.566"></a>
<FONT color="green">567</FONT>    <a name="line.567"></a>
<FONT color="green">568</FONT>            final int m     = main.length;<a name="line.568"></a>
<FONT color="green">569</FONT>            final int lowerStart = 4 * m;<a name="line.569"></a>
<FONT color="green">570</FONT>            final int upperStart = 5 * m;<a name="line.570"></a>
<FONT color="green">571</FONT>            lowerSpectra = Double.POSITIVE_INFINITY;<a name="line.571"></a>
<FONT color="green">572</FONT>            upperSpectra = Double.NEGATIVE_INFINITY;<a name="line.572"></a>
<FONT color="green">573</FONT>            double eMax = 0;<a name="line.573"></a>
<FONT color="green">574</FONT>    <a name="line.574"></a>
<FONT color="green">575</FONT>            double eCurrent = 0;<a name="line.575"></a>
<FONT color="green">576</FONT>            for (int i = 0; i &lt; m - 1; ++i) {<a name="line.576"></a>
<FONT color="green">577</FONT>    <a name="line.577"></a>
<FONT color="green">578</FONT>                final double dCurrent = main[i];<a name="line.578"></a>
<FONT color="green">579</FONT>                final double ePrevious = eCurrent;<a name="line.579"></a>
<FONT color="green">580</FONT>                eCurrent = Math.abs(secondary[i]);<a name="line.580"></a>
<FONT color="green">581</FONT>                eMax = Math.max(eMax, eCurrent);<a name="line.581"></a>
<FONT color="green">582</FONT>                final double radius = ePrevious + eCurrent;<a name="line.582"></a>
<FONT color="green">583</FONT>    <a name="line.583"></a>
<FONT color="green">584</FONT>                final double lower = dCurrent - radius;<a name="line.584"></a>
<FONT color="green">585</FONT>                work[lowerStart + i] = lower;<a name="line.585"></a>
<FONT color="green">586</FONT>                lowerSpectra = Math.min(lowerSpectra, lower);<a name="line.586"></a>
<FONT color="green">587</FONT>    <a name="line.587"></a>
<FONT color="green">588</FONT>                final double upper = dCurrent + radius;<a name="line.588"></a>
<FONT color="green">589</FONT>                work[upperStart + i] = upper;<a name="line.589"></a>
<FONT color="green">590</FONT>                upperSpectra = Math.max(upperSpectra, upper);<a name="line.590"></a>
<FONT color="green">591</FONT>                <a name="line.591"></a>
<FONT color="green">592</FONT>            }<a name="line.592"></a>
<FONT color="green">593</FONT>    <a name="line.593"></a>
<FONT color="green">594</FONT>            final double dCurrent = main[m - 1];<a name="line.594"></a>
<FONT color="green">595</FONT>            work[lowerStart + m - 1] = dCurrent - eCurrent;<a name="line.595"></a>
<FONT color="green">596</FONT>            work[upperStart + m - 1] = dCurrent + eCurrent;<a name="line.596"></a>
<FONT color="green">597</FONT>            minPivot = MathUtils.SAFE_MIN * Math.max(1.0, eMax * eMax);<a name="line.597"></a>
<FONT color="green">598</FONT>    <a name="line.598"></a>
<FONT color="green">599</FONT>        }<a name="line.599"></a>
<FONT color="green">600</FONT>    <a name="line.600"></a>
<FONT color="green">601</FONT>        /**<a name="line.601"></a>
<FONT color="green">602</FONT>         * Find the realEigenvalues.<a name="line.602"></a>
<FONT color="green">603</FONT>         * @exception InvalidMatrixException if a block cannot be diagonalized<a name="line.603"></a>
<FONT color="green">604</FONT>         */<a name="line.604"></a>
<FONT color="green">605</FONT>        private void findEigenvalues()<a name="line.605"></a>
<FONT color="green">606</FONT>            throws InvalidMatrixException {<a name="line.606"></a>
<FONT color="green">607</FONT>    <a name="line.607"></a>
<FONT color="green">608</FONT>            // compute splitting points<a name="line.608"></a>
<FONT color="green">609</FONT>            List&lt;Integer&gt; splitIndices = computeSplits();<a name="line.609"></a>
<FONT color="green">610</FONT>    <a name="line.610"></a>
<FONT color="green">611</FONT>            // find realEigenvalues in each block<a name="line.611"></a>
<FONT color="green">612</FONT>            realEigenvalues = new double[main.length];<a name="line.612"></a>
<FONT color="green">613</FONT>            imagEigenvalues = new double[main.length];<a name="line.613"></a>
<FONT color="green">614</FONT>            int begin = 0;<a name="line.614"></a>
<FONT color="green">615</FONT>            for (final int end : splitIndices) {<a name="line.615"></a>
<FONT color="green">616</FONT>                final int n = end - begin;<a name="line.616"></a>
<FONT color="green">617</FONT>                switch (n) {<a name="line.617"></a>
<FONT color="green">618</FONT>    <a name="line.618"></a>
<FONT color="green">619</FONT>                case 1:<a name="line.619"></a>
<FONT color="green">620</FONT>                    // apply dedicated method for dimension 1<a name="line.620"></a>
<FONT color="green">621</FONT>                    process1RowBlock(begin);<a name="line.621"></a>
<FONT color="green">622</FONT>                    break;<a name="line.622"></a>
<FONT color="green">623</FONT>    <a name="line.623"></a>
<FONT color="green">624</FONT>                case 2:<a name="line.624"></a>
<FONT color="green">625</FONT>                    // apply dedicated method for dimension 2<a name="line.625"></a>
<FONT color="green">626</FONT>                    process2RowsBlock(begin);<a name="line.626"></a>
<FONT color="green">627</FONT>                    break;<a name="line.627"></a>
<FONT color="green">628</FONT>    <a name="line.628"></a>
<FONT color="green">629</FONT>                case 3:<a name="line.629"></a>
<FONT color="green">630</FONT>                    // apply dedicated method for dimension 3<a name="line.630"></a>
<FONT color="green">631</FONT>                    process3RowsBlock(begin);<a name="line.631"></a>
<FONT color="green">632</FONT>                    break;<a name="line.632"></a>
<FONT color="green">633</FONT>    <a name="line.633"></a>
<FONT color="green">634</FONT>                default:<a name="line.634"></a>
<FONT color="green">635</FONT>    <a name="line.635"></a>
<FONT color="green">636</FONT>                    // choose an initial shift for LDL&lt;sup&gt;T&lt;/sup&gt; decomposition<a name="line.636"></a>
<FONT color="green">637</FONT>                    final double[] range       = eigenvaluesRange(begin, n);<a name="line.637"></a>
<FONT color="green">638</FONT>                    final double oneFourth     = 0.25 * (3 * range[0] + range[1]);<a name="line.638"></a>
<FONT color="green">639</FONT>                    final int oneFourthCount   = countEigenValues(oneFourth, begin, n);<a name="line.639"></a>
<FONT color="green">640</FONT>                    final double threeFourth   = 0.25 * (range[0] + 3 * range[1]);<a name="line.640"></a>
<FONT color="green">641</FONT>                    final int threeFourthCount = countEigenValues(threeFourth, begin, n);<a name="line.641"></a>
<FONT color="green">642</FONT>                    final boolean chooseLeft   = (oneFourthCount - 1) &gt;= (n - threeFourthCount);<a name="line.642"></a>
<FONT color="green">643</FONT>                    final double lambda        = chooseLeft ? range[0] : range[1];<a name="line.643"></a>
<FONT color="green">644</FONT>    <a name="line.644"></a>
<FONT color="green">645</FONT>                    tau = (range[1] - range[0]) * MathUtils.EPSILON * n + 2 * minPivot;<a name="line.645"></a>
<FONT color="green">646</FONT>    <a name="line.646"></a>
<FONT color="green">647</FONT>                    // decompose T&amp;lambda;I as LDL&lt;sup&gt;T&lt;/sup&gt;<a name="line.647"></a>
<FONT color="green">648</FONT>                    ldlTDecomposition(lambda, begin, n);<a name="line.648"></a>
<FONT color="green">649</FONT>    <a name="line.649"></a>
<FONT color="green">650</FONT>                    // apply general dqd/dqds method<a name="line.650"></a>
<FONT color="green">651</FONT>                    processGeneralBlock(n);<a name="line.651"></a>
<FONT color="green">652</FONT>    <a name="line.652"></a>
<FONT color="green">653</FONT>                    // extract realEigenvalues<a name="line.653"></a>
<FONT color="green">654</FONT>                    if (chooseLeft) {<a name="line.654"></a>
<FONT color="green">655</FONT>                        for (int i = 0; i &lt; n; ++i) {<a name="line.655"></a>
<FONT color="green">656</FONT>                            realEigenvalues[begin + i] = lambda + work[4 * i];<a name="line.656"></a>
<FONT color="green">657</FONT>                        }<a name="line.657"></a>
<FONT color="green">658</FONT>                    } else {<a name="line.658"></a>
<FONT color="green">659</FONT>                        for (int i = 0; i &lt; n; ++i) {<a name="line.659"></a>
<FONT color="green">660</FONT>                            realEigenvalues[begin + i] = lambda - work[4 * i];<a name="line.660"></a>
<FONT color="green">661</FONT>                        }                    <a name="line.661"></a>
<FONT color="green">662</FONT>                    }<a name="line.662"></a>
<FONT color="green">663</FONT>    <a name="line.663"></a>
<FONT color="green">664</FONT>                }<a name="line.664"></a>
<FONT color="green">665</FONT>                begin = end;<a name="line.665"></a>
<FONT color="green">666</FONT>            }<a name="line.666"></a>
<FONT color="green">667</FONT>    <a name="line.667"></a>
<FONT color="green">668</FONT>            // sort the realEigenvalues in decreasing order<a name="line.668"></a>
<FONT color="green">669</FONT>            Arrays.sort(realEigenvalues);<a name="line.669"></a>
<FONT color="green">670</FONT>            for (int i = 0, j = realEigenvalues.length - 1; i &lt; j; ++i, --j) {<a name="line.670"></a>
<FONT color="green">671</FONT>                final double tmp = realEigenvalues[i];<a name="line.671"></a>
<FONT color="green">672</FONT>                realEigenvalues[i] = realEigenvalues[j];<a name="line.672"></a>
<FONT color="green">673</FONT>                realEigenvalues[j] = tmp;<a name="line.673"></a>
<FONT color="green">674</FONT>            }<a name="line.674"></a>
<FONT color="green">675</FONT>    <a name="line.675"></a>
<FONT color="green">676</FONT>        }<a name="line.676"></a>
<FONT color="green">677</FONT>    <a name="line.677"></a>
<FONT color="green">678</FONT>        /**<a name="line.678"></a>
<FONT color="green">679</FONT>         * Compute splitting points.<a name="line.679"></a>
<FONT color="green">680</FONT>         * @return list of indices after matrix can be split<a name="line.680"></a>
<FONT color="green">681</FONT>         */<a name="line.681"></a>
<FONT color="green">682</FONT>        private List&lt;Integer&gt; computeSplits() {<a name="line.682"></a>
<FONT color="green">683</FONT>    <a name="line.683"></a>
<FONT color="green">684</FONT>            final List&lt;Integer&gt; list = new ArrayList&lt;Integer&gt;();<a name="line.684"></a>
<FONT color="green">685</FONT>    <a name="line.685"></a>
<FONT color="green">686</FONT>            // splitting preserving relative accuracy<a name="line.686"></a>
<FONT color="green">687</FONT>            double absDCurrent = Math.abs(main[0]);<a name="line.687"></a>
<FONT color="green">688</FONT>            for (int i = 0; i &lt; secondary.length; ++i) {<a name="line.688"></a>
<FONT color="green">689</FONT>                final double absDPrevious = absDCurrent;<a name="line.689"></a>
<FONT color="green">690</FONT>                absDCurrent = Math.abs(main[i + 1]);<a name="line.690"></a>
<FONT color="green">691</FONT>                final double max = splitTolerance * Math.sqrt(absDPrevious * absDCurrent);<a name="line.691"></a>
<FONT color="green">692</FONT>                if (Math.abs(secondary[i]) &lt;= max) {<a name="line.692"></a>
<FONT color="green">693</FONT>                    list.add(i + 1);<a name="line.693"></a>
<FONT color="green">694</FONT>                    secondary[i] = 0;<a name="line.694"></a>
<FONT color="green">695</FONT>                    squaredSecondary[i] = 0;<a name="line.695"></a>
<FONT color="green">696</FONT>                }<a name="line.696"></a>
<FONT color="green">697</FONT>            }<a name="line.697"></a>
<FONT color="green">698</FONT>    <a name="line.698"></a>
<FONT color="green">699</FONT>            list.add(secondary.length + 1);<a name="line.699"></a>
<FONT color="green">700</FONT>            return list;<a name="line.700"></a>
<FONT color="green">701</FONT>    <a name="line.701"></a>
<FONT color="green">702</FONT>        }<a name="line.702"></a>
<FONT color="green">703</FONT>    <a name="line.703"></a>
<FONT color="green">704</FONT>        /**<a name="line.704"></a>
<FONT color="green">705</FONT>         * Find eigenvalue in a block with 1 row.<a name="line.705"></a>
<FONT color="green">706</FONT>         * &lt;p&gt;In low dimensions, we simply solve the characteristic polynomial.&lt;/p&gt;<a name="line.706"></a>
<FONT color="green">707</FONT>         * @param index index of the first row of the block<a name="line.707"></a>
<FONT color="green">708</FONT>         */<a name="line.708"></a>
<FONT color="green">709</FONT>        private void process1RowBlock(final int index) {<a name="line.709"></a>
<FONT color="green">710</FONT>            realEigenvalues[index] = main[index];<a name="line.710"></a>
<FONT color="green">711</FONT>        }<a name="line.711"></a>
<FONT color="green">712</FONT>    <a name="line.712"></a>
<FONT color="green">713</FONT>        /**<a name="line.713"></a>
<FONT color="green">714</FONT>         * Find realEigenvalues in a block with 2 rows.<a name="line.714"></a>
<FONT color="green">715</FONT>         * &lt;p&gt;In low dimensions, we simply solve the characteristic polynomial.&lt;/p&gt;<a name="line.715"></a>
<FONT color="green">716</FONT>         * @param index index of the first row of the block<a name="line.716"></a>
<FONT color="green">717</FONT>         * @exception InvalidMatrixException if characteristic polynomial cannot be solved<a name="line.717"></a>
<FONT color="green">718</FONT>         */<a name="line.718"></a>
<FONT color="green">719</FONT>        private void process2RowsBlock(final int index)<a name="line.719"></a>
<FONT color="green">720</FONT>            throws InvalidMatrixException {<a name="line.720"></a>
<FONT color="green">721</FONT>    <a name="line.721"></a>
<FONT color="green">722</FONT>            // the characteristic polynomial is<a name="line.722"></a>
<FONT color="green">723</FONT>            // X^2 - (q0 + q1) X + q0 q1 - e1^2<a name="line.723"></a>
<FONT color="green">724</FONT>            final double q0   = main[index];<a name="line.724"></a>
<FONT color="green">725</FONT>            final double q1   = main[index + 1];<a name="line.725"></a>
<FONT color="green">726</FONT>            final double e12  = squaredSecondary[index];<a name="line.726"></a>
<FONT color="green">727</FONT>    <a name="line.727"></a>
<FONT color="green">728</FONT>            final double s     = q0 + q1;<a name="line.728"></a>
<FONT color="green">729</FONT>            final double p     = q0 * q1 - e12;<a name="line.729"></a>
<FONT color="green">730</FONT>            final double delta = s * s - 4 * p;<a name="line.730"></a>
<FONT color="green">731</FONT>            if (delta &lt; 0) {<a name="line.731"></a>
<FONT color="green">732</FONT>                throw new InvalidMatrixException("cannot solve degree {0} equation", 2);<a name="line.732"></a>
<FONT color="green">733</FONT>            }<a name="line.733"></a>
<FONT color="green">734</FONT>    <a name="line.734"></a>
<FONT color="green">735</FONT>            final double largestRoot = 0.5 * (s + Math.sqrt(delta));<a name="line.735"></a>
<FONT color="green">736</FONT>            realEigenvalues[index]     = largestRoot;<a name="line.736"></a>
<FONT color="green">737</FONT>            realEigenvalues[index + 1] = p / largestRoot;<a name="line.737"></a>
<FONT color="green">738</FONT>    <a name="line.738"></a>
<FONT color="green">739</FONT>        }<a name="line.739"></a>
<FONT color="green">740</FONT>    <a name="line.740"></a>
<FONT color="green">741</FONT>        /**<a name="line.741"></a>
<FONT color="green">742</FONT>         * Find realEigenvalues in a block with 3 rows.<a name="line.742"></a>
<FONT color="green">743</FONT>         * &lt;p&gt;In low dimensions, we simply solve the characteristic polynomial.&lt;/p&gt;<a name="line.743"></a>
<FONT color="green">744</FONT>         * @param index index of the first row of the block<a name="line.744"></a>
<FONT color="green">745</FONT>         * @exception InvalidMatrixException if diagonal elements are not positive<a name="line.745"></a>
<FONT color="green">746</FONT>         */<a name="line.746"></a>
<FONT color="green">747</FONT>        private void process3RowsBlock(final int index)<a name="line.747"></a>
<FONT color="green">748</FONT>            throws InvalidMatrixException {<a name="line.748"></a>
<FONT color="green">749</FONT>    <a name="line.749"></a>
<FONT color="green">750</FONT>            // the characteristic polynomial is<a name="line.750"></a>
<FONT color="green">751</FONT>            // X^3 - (q0 + q1 + q2) X^2 + (q0 q1 + q0 q2 + q1 q2 - e1^2 - e2^2) X + q0 e2^2 + q2 e1^2 - q0 q1 q2<a name="line.751"></a>
<FONT color="green">752</FONT>            final double q0       = main[index];<a name="line.752"></a>
<FONT color="green">753</FONT>            final double q1       = main[index + 1];<a name="line.753"></a>
<FONT color="green">754</FONT>            final double q2       = main[index + 2];<a name="line.754"></a>
<FONT color="green">755</FONT>            final double e12      = squaredSecondary[index];<a name="line.755"></a>
<FONT color="green">756</FONT>            final double q1q2Me22 = q1 * q2 - squaredSecondary[index + 1];<a name="line.756"></a>
<FONT color="green">757</FONT>    <a name="line.757"></a>
<FONT color="green">758</FONT>            // compute coefficients of the cubic equation as: x^3 + b x^2 + c x + d = 0<a name="line.758"></a>
<FONT color="green">759</FONT>            final double b        = -(q0 + q1 + q2);<a name="line.759"></a>
<FONT color="green">760</FONT>            final double c        = q0 * q1 + q0 * q2 + q1q2Me22 - e12;<a name="line.760"></a>
<FONT color="green">761</FONT>            final double d        = q2 * e12 - q0 * q1q2Me22;<a name="line.761"></a>
<FONT color="green">762</FONT>    <a name="line.762"></a>
<FONT color="green">763</FONT>            // solve cubic equation<a name="line.763"></a>
<FONT color="green">764</FONT>            final double b2       = b * b;<a name="line.764"></a>
<FONT color="green">765</FONT>            final double q        = (3 * c - b2) / 9;<a name="line.765"></a>
<FONT color="green">766</FONT>            final double r        = ((9 * c - 2 * b2) * b - 27 * d) / 54;<a name="line.766"></a>
<FONT color="green">767</FONT>            final double delta    = q * q * q + r * r;<a name="line.767"></a>
<FONT color="green">768</FONT>            if (delta &gt;= 0) {<a name="line.768"></a>
<FONT color="green">769</FONT>                // in fact, there are solutions to the equation, but in the context<a name="line.769"></a>
<FONT color="green">770</FONT>                // of symmetric realEigenvalues problem, there should be three distinct<a name="line.770"></a>
<FONT color="green">771</FONT>                // real roots, so we throw an error if this condition is not met<a name="line.771"></a>
<FONT color="green">772</FONT>                throw new InvalidMatrixException("cannot solve degree {0} equation", 3);           <a name="line.772"></a>
<FONT color="green">773</FONT>            }<a name="line.773"></a>
<FONT color="green">774</FONT>            final double sqrtMq = Math.sqrt(-q);<a name="line.774"></a>
<FONT color="green">775</FONT>            final double theta  = Math.acos(r / (-q * sqrtMq));<a name="line.775"></a>
<FONT color="green">776</FONT>            final double alpha  = 2 * sqrtMq;<a name="line.776"></a>
<FONT color="green">777</FONT>            final double beta   = b / 3;<a name="line.777"></a>
<FONT color="green">778</FONT>    <a name="line.778"></a>
<FONT color="green">779</FONT>            double z0 = alpha * Math.cos(theta / 3) - beta;<a name="line.779"></a>
<FONT color="green">780</FONT>            double z1 = alpha * Math.cos((theta + 2 * Math.PI) / 3) - beta;<a name="line.780"></a>
<FONT color="green">781</FONT>            double z2 = alpha * Math.cos((theta + 4 * Math.PI) / 3) - beta;<a name="line.781"></a>
<FONT color="green">782</FONT>            if (z0 &lt; z1) {<a name="line.782"></a>
<FONT color="green">783</FONT>                final double t = z0;<a name="line.783"></a>
<FONT color="green">784</FONT>                z0 = z1;<a name="line.784"></a>
<FONT color="green">785</FONT>                z1 = t;<a name="line.785"></a>
<FONT color="green">786</FONT>            }<a name="line.786"></a>
<FONT color="green">787</FONT>            if (z1 &lt; z2) {<a name="line.787"></a>
<FONT color="green">788</FONT>                final double t = z1;<a name="line.788"></a>
<FONT color="green">789</FONT>                z1 = z2;<a name="line.789"></a>
<FONT color="green">790</FONT>                z2 = t;<a name="line.790"></a>
<FONT color="green">791</FONT>            }<a name="line.791"></a>
<FONT color="green">792</FONT>            if (z0 &lt; z1) {<a name="line.792"></a>
<FONT color="green">793</FONT>                final double t = z0;<a name="line.793"></a>
<FONT color="green">794</FONT>                z0 = z1;<a name="line.794"></a>
<FONT color="green">795</FONT>                z1 = t;<a name="line.795"></a>
<FONT color="green">796</FONT>            }<a name="line.796"></a>
<FONT color="green">797</FONT>            realEigenvalues[index]     = z0;<a name="line.797"></a>
<FONT color="green">798</FONT>            realEigenvalues[index + 1] = z1;<a name="line.798"></a>
<FONT color="green">799</FONT>            realEigenvalues[index + 2] = z2;<a name="line.799"></a>
<FONT color="green">800</FONT>    <a name="line.800"></a>
<FONT color="green">801</FONT>        }<a name="line.801"></a>
<FONT color="green">802</FONT>    <a name="line.802"></a>
<FONT color="green">803</FONT>        /**<a name="line.803"></a>
<FONT color="green">804</FONT>         * Find realEigenvalues using dqd/dqds algorithms.<a name="line.804"></a>
<FONT color="green">805</FONT>         * &lt;p&gt;This implementation is based on Beresford N. Parlett<a name="line.805"></a>
<FONT color="green">806</FONT>         * and Osni A. Marques paper &lt;a<a name="line.806"></a>
<FONT color="green">807</FONT>         * href="http://www.netlib.org/lapack/lawnspdf/lawn155.pdf"&gt;An<a name="line.807"></a>
<FONT color="green">808</FONT>         * Implementation of the dqds Algorithm (Positive Case)&lt;/a&gt; and on the<a name="line.808"></a>
<FONT color="green">809</FONT>         * corresponding LAPACK routine DLASQ2.&lt;/p&gt;<a name="line.809"></a>
<FONT color="green">810</FONT>         * @param n number of rows of the block<a name="line.810"></a>
<FONT color="green">811</FONT>         * @exception InvalidMatrixException if block cannot be diagonalized<a name="line.811"></a>
<FONT color="green">812</FONT>         * after 30 * n iterations<a name="line.812"></a>
<FONT color="green">813</FONT>         */<a name="line.813"></a>
<FONT color="green">814</FONT>        private void processGeneralBlock(final int n)<a name="line.814"></a>
<FONT color="green">815</FONT>            throws InvalidMatrixException {<a name="line.815"></a>
<FONT color="green">816</FONT>    <a name="line.816"></a>
<FONT color="green">817</FONT>            // check decomposed matrix data range<a name="line.817"></a>
<FONT color="green">818</FONT>            double sumOffDiag = 0;<a name="line.818"></a>
<FONT color="green">819</FONT>            for (int i = 0; i &lt; n - 1; ++i) {<a name="line.819"></a>
<FONT color="green">820</FONT>                final int fourI = 4 * i;<a name="line.820"></a>
<FONT color="green">821</FONT>                final double ei = work[fourI + 2];<a name="line.821"></a>
<FONT color="green">822</FONT>                sumOffDiag += ei;<a name="line.822"></a>
<FONT color="green">823</FONT>            }<a name="line.823"></a>
<FONT color="green">824</FONT>    <a name="line.824"></a>
<FONT color="green">825</FONT>            if (sumOffDiag == 0) {<a name="line.825"></a>
<FONT color="green">826</FONT>                // matrix is already diagonal<a name="line.826"></a>
<FONT color="green">827</FONT>                return;<a name="line.827"></a>
<FONT color="green">828</FONT>            }<a name="line.828"></a>
<FONT color="green">829</FONT>    <a name="line.829"></a>
<FONT color="green">830</FONT>            // initial checks for splits (see Parlett &amp; Marques section 3.3)<a name="line.830"></a>
<FONT color="green">831</FONT>            flipIfWarranted(n, 2);<a name="line.831"></a>
<FONT color="green">832</FONT>    <a name="line.832"></a>
<FONT color="green">833</FONT>            // two iterations with Li's test for initial splits<a name="line.833"></a>
<FONT color="green">834</FONT>            initialSplits(n);<a name="line.834"></a>
<FONT color="green">835</FONT>    <a name="line.835"></a>
<FONT color="green">836</FONT>            // initialize parameters used by goodStep<a name="line.836"></a>
<FONT color="green">837</FONT>            tType = 0;<a name="line.837"></a>
<FONT color="green">838</FONT>            dMin1 = 0;<a name="line.838"></a>
<FONT color="green">839</FONT>            dMin2 = 0;<a name="line.839"></a>
<FONT color="green">840</FONT>            dN    = 0;<a name="line.840"></a>
<FONT color="green">841</FONT>            dN1   = 0;<a name="line.841"></a>
<FONT color="green">842</FONT>            dN2   = 0;<a name="line.842"></a>
<FONT color="green">843</FONT>            tau   = 0;<a name="line.843"></a>
<FONT color="green">844</FONT>    <a name="line.844"></a>
<FONT color="green">845</FONT>            // process split segments<a name="line.845"></a>
<FONT color="green">846</FONT>            int i0 = 0;<a name="line.846"></a>
<FONT color="green">847</FONT>            int n0 = n;<a name="line.847"></a>
<FONT color="green">848</FONT>            while (n0 &gt; 0) {<a name="line.848"></a>
<FONT color="green">849</FONT>    <a name="line.849"></a>
<FONT color="green">850</FONT>                // retrieve shift that was temporarily stored as a negative off-diagonal element<a name="line.850"></a>
<FONT color="green">851</FONT>                sigma    = (n0 == n) ? 0 : -work[4 * n0 - 2];<a name="line.851"></a>
<FONT color="green">852</FONT>                sigmaLow = 0;<a name="line.852"></a>
<FONT color="green">853</FONT>    <a name="line.853"></a>
<FONT color="green">854</FONT>                // find start of a new split segment to process<a name="line.854"></a>
<FONT color="green">855</FONT>                double eMin = (i0 == n0) ? 0 : work[4 * n0 - 6];<a name="line.855"></a>
<FONT color="green">856</FONT>                double eMax = 0;<a name="line.856"></a>
<FONT color="green">857</FONT>                double qMax = work[4 * n0 - 4];<a name="line.857"></a>
<FONT color="green">858</FONT>                double qMin = qMax;<a name="line.858"></a>
<FONT color="green">859</FONT>                i0 = 0;<a name="line.859"></a>
<FONT color="green">860</FONT>                for (int i = 4 * (n0 - 2); i &gt;= 0; i -= 4) {<a name="line.860"></a>
<FONT color="green">861</FONT>                    if (work[i + 2] &lt;= 0) {<a name="line.861"></a>
<FONT color="green">862</FONT>                        i0 = 1 + i / 4;<a name="line.862"></a>
<FONT color="green">863</FONT>                        break;<a name="line.863"></a>
<FONT color="green">864</FONT>                    }<a name="line.864"></a>
<FONT color="green">865</FONT>                    if (qMin &gt;= 4 * eMax) {<a name="line.865"></a>
<FONT color="green">866</FONT>                        qMin = Math.min(qMin, work[i + 4]);<a name="line.866"></a>
<FONT color="green">867</FONT>                        eMax = Math.max(eMax, work[i + 2]);<a name="line.867"></a>
<FONT color="green">868</FONT>                    }<a name="line.868"></a>
<FONT color="green">869</FONT>                    qMax = Math.max(qMax, work[i] + work[i + 2]);<a name="line.869"></a>
<FONT color="green">870</FONT>                    eMin = Math.min(eMin, work[i + 2]);<a name="line.870"></a>
<FONT color="green">871</FONT>                }<a name="line.871"></a>
<FONT color="green">872</FONT>                work[4 * n0 - 2] = eMin;<a name="line.872"></a>
<FONT color="green">873</FONT>    <a name="line.873"></a>
<FONT color="green">874</FONT>                // lower bound of Gershgorin disk<a name="line.874"></a>
<FONT color="green">875</FONT>                dMin = -Math.max(0, qMin - 2 * Math.sqrt(qMin * eMax));<a name="line.875"></a>
<FONT color="green">876</FONT>    <a name="line.876"></a>
<FONT color="green">877</FONT>                pingPong = 0;<a name="line.877"></a>
<FONT color="green">878</FONT>                int maxIter = 30 * (n0 - i0);<a name="line.878"></a>
<FONT color="green">879</FONT>                for (int k = 0; i0 &lt; n0; ++k) {<a name="line.879"></a>
<FONT color="green">880</FONT>                    if (k &gt;= maxIter) {<a name="line.880"></a>
<FONT color="green">881</FONT>                        throw new InvalidMatrixException(new MaxIterationsExceededException(maxIter));<a name="line.881"></a>
<FONT color="green">882</FONT>                    }<a name="line.882"></a>
<FONT color="green">883</FONT>    <a name="line.883"></a>
<FONT color="green">884</FONT>                    // perform one step<a name="line.884"></a>
<FONT color="green">885</FONT>                    n0 = goodStep(i0, n0);<a name="line.885"></a>
<FONT color="green">886</FONT>                    pingPong = 1 - pingPong;<a name="line.886"></a>
<FONT color="green">887</FONT>    <a name="line.887"></a>
<FONT color="green">888</FONT>                    // check for new splits after "ping" steps<a name="line.888"></a>
<FONT color="green">889</FONT>                    // when the last elements of qd array are very small<a name="line.889"></a>
<FONT color="green">890</FONT>                    if ((pingPong == 0) &amp;&amp; (n0 - i0 &gt; 3) &amp;&amp;<a name="line.890"></a>
<FONT color="green">891</FONT>                        (work[4 * n0 - 1] &lt;= TOLERANCE_2 * qMax) &amp;&amp;<a name="line.891"></a>
<FONT color="green">892</FONT>                        (work[4 * n0 - 2] &lt;= TOLERANCE_2 * sigma)) {<a name="line.892"></a>
<FONT color="green">893</FONT>                        int split = i0 - 1;<a name="line.893"></a>
<FONT color="green">894</FONT>                        qMax = work[4 * i0];<a name="line.894"></a>
<FONT color="green">895</FONT>                        eMin = work[4 * i0 + 2];<a name="line.895"></a>
<FONT color="green">896</FONT>                        double previousEMin = work[4 * i0 + 3];<a name="line.896"></a>
<FONT color="green">897</FONT>                        for (int i = 4 * i0; i &lt; 4 * n0 - 11; i += 4) {<a name="line.897"></a>
<FONT color="green">898</FONT>                            if ((work[i + 3] &lt;= TOLERANCE_2 * work[i]) &amp;&amp;<a name="line.898"></a>
<FONT color="green">899</FONT>                                (work[i + 2] &lt;= TOLERANCE_2 * sigma)) {<a name="line.899"></a>
<FONT color="green">900</FONT>                                // insert a split<a name="line.900"></a>
<FONT color="green">901</FONT>                                work[i + 2]  = -sigma;<a name="line.901"></a>
<FONT color="green">902</FONT>                                split        = i / 4;<a name="line.902"></a>
<FONT color="green">903</FONT>                                qMax         = 0;<a name="line.903"></a>
<FONT color="green">904</FONT>                                eMin         = work[i + 6];<a name="line.904"></a>
<FONT color="green">905</FONT>                                previousEMin = work[i + 7];<a name="line.905"></a>
<FONT color="green">906</FONT>                            } else {<a name="line.906"></a>
<FONT color="green">907</FONT>                                qMax         = Math.max(qMax, work[i + 4]);<a name="line.907"></a>
<FONT color="green">908</FONT>                                eMin         = Math.min(eMin, work[i + 2]);<a name="line.908"></a>
<FONT color="green">909</FONT>                                previousEMin = Math.min(previousEMin, work[i + 3]);<a name="line.909"></a>
<FONT color="green">910</FONT>                            }<a name="line.910"></a>
<FONT color="green">911</FONT>                        }<a name="line.911"></a>
<FONT color="green">912</FONT>                        work[4 * n0 - 2] = eMin;<a name="line.912"></a>
<FONT color="green">913</FONT>                        work[4 * n0 - 1] = previousEMin;<a name="line.913"></a>
<FONT color="green">914</FONT>                        i0 = split + 1;<a name="line.914"></a>
<FONT color="green">915</FONT>                    }<a name="line.915"></a>
<FONT color="green">916</FONT>                }<a name="line.916"></a>
<FONT color="green">917</FONT>    <a name="line.917"></a>
<FONT color="green">918</FONT>            }<a name="line.918"></a>
<FONT color="green">919</FONT>    <a name="line.919"></a>
<FONT color="green">920</FONT>        }<a name="line.920"></a>
<FONT color="green">921</FONT>    <a name="line.921"></a>
<FONT color="green">922</FONT>        /**<a name="line.922"></a>
<FONT color="green">923</FONT>         * Perform two iterations with Li's tests for initial splits.<a name="line.923"></a>
<FONT color="green">924</FONT>         * @param n number of rows of the matrix to process<a name="line.924"></a>
<FONT color="green">925</FONT>         */<a name="line.925"></a>
<FONT color="green">926</FONT>        private void initialSplits(final int n) {<a name="line.926"></a>
<FONT color="green">927</FONT>    <a name="line.927"></a>
<FONT color="green">928</FONT>            pingPong = 0;<a name="line.928"></a>
<FONT color="green">929</FONT>            for (int k = 0; k &lt; 2; ++k) {<a name="line.929"></a>
<FONT color="green">930</FONT>    <a name="line.930"></a>
<FONT color="green">931</FONT>                // apply Li's reverse test<a name="line.931"></a>
<FONT color="green">932</FONT>                double d = work[4 * (n - 1) + pingPong];<a name="line.932"></a>
<FONT color="green">933</FONT>                for (int i = 4 * (n - 2) + pingPong; i &gt;= 0; i -= 4) {<a name="line.933"></a>
<FONT color="green">934</FONT>                    if (work[i + 2] &lt;= TOLERANCE_2 * d) {<a name="line.934"></a>
<FONT color="green">935</FONT>                        work[i + 2] = -0.0;<a name="line.935"></a>
<FONT color="green">936</FONT>                        d = work[i];<a name="line.936"></a>
<FONT color="green">937</FONT>                    } else {<a name="line.937"></a>
<FONT color="green">938</FONT>                        d *= work[i] / (d + work[i + 2]);<a name="line.938"></a>
<FONT color="green">939</FONT>                    }<a name="line.939"></a>
<FONT color="green">940</FONT>                }<a name="line.940"></a>
<FONT color="green">941</FONT>    <a name="line.941"></a>
<FONT color="green">942</FONT>                // apply dqd plus Li's forward test.<a name="line.942"></a>
<FONT color="green">943</FONT>                d = work[pingPong];<a name="line.943"></a>
<FONT color="green">944</FONT>                for (int i = 2 + pingPong; i &lt; 4 * n - 2; i += 4) {<a name="line.944"></a>
<FONT color="green">945</FONT>                    final int j = i - 2 * pingPong - 1;<a name="line.945"></a>
<FONT color="green">946</FONT>                    work[j] = d + work[i];<a name="line.946"></a>
<FONT color="green">947</FONT>                    if (work[i] &lt;= TOLERANCE_2 * d) {<a name="line.947"></a>
<FONT color="green">948</FONT>                        work[i]     = -0.0;<a name="line.948"></a>
<FONT color="green">949</FONT>                        work[j]     = d;<a name="line.949"></a>
<FONT color="green">950</FONT>                        work[j + 2] = 0.0;<a name="line.950"></a>
<FONT color="green">951</FONT>                        d = work[i + 2];<a name="line.951"></a>
<FONT color="green">952</FONT>                    } else if ((MathUtils.SAFE_MIN * work[i + 2] &lt; work[j]) &amp;&amp;<a name="line.952"></a>
<FONT color="green">953</FONT>                               (MathUtils.SAFE_MIN * work[j] &lt; work[i + 2])) {<a name="line.953"></a>
<FONT color="green">954</FONT>                        final double tmp = work[i + 2] / work[j];<a name="line.954"></a>
<FONT color="green">955</FONT>                        work[j + 2] = work[i] * tmp;<a name="line.955"></a>
<FONT color="green">956</FONT>                        d *= tmp;<a name="line.956"></a>
<FONT color="green">957</FONT>                    } else {<a name="line.957"></a>
<FONT color="green">958</FONT>                        work[j + 2] = work[i + 2] * (work[i] / work[j]);<a name="line.958"></a>
<FONT color="green">959</FONT>                        d *= work[i + 2] / work[j];<a name="line.959"></a>
<FONT color="green">960</FONT>                   }<a name="line.960"></a>
<FONT color="green">961</FONT>                }<a name="line.961"></a>
<FONT color="green">962</FONT>                work[4 * n - 3 - pingPong] = d;<a name="line.962"></a>
<FONT color="green">963</FONT>    <a name="line.963"></a>
<FONT color="green">964</FONT>                // from ping to pong<a name="line.964"></a>
<FONT color="green">965</FONT>                pingPong = 1 - pingPong;<a name="line.965"></a>
<FONT color="green">966</FONT>    <a name="line.966"></a>
<FONT color="green">967</FONT>            }<a name="line.967"></a>
<FONT color="green">968</FONT>    <a name="line.968"></a>
<FONT color="green">969</FONT>        }<a name="line.969"></a>
<FONT color="green">970</FONT>    <a name="line.970"></a>
<FONT color="green">971</FONT>        /**<a name="line.971"></a>
<FONT color="green">972</FONT>         * Perform one "good" dqd/dqds step.<a name="line.972"></a>
<FONT color="green">973</FONT>         * &lt;p&gt;This implementation is based on Beresford N. Parlett<a name="line.973"></a>
<FONT color="green">974</FONT>         * and Osni A. Marques paper &lt;a<a name="line.974"></a>
<FONT color="green">975</FONT>         * href="http://www.netlib.org/lapack/lawnspdf/lawn155.pdf"&gt;An<a name="line.975"></a>
<FONT color="green">976</FONT>         * Implementation of the dqds Algorithm (Positive Case)&lt;/a&gt; and on the<a name="line.976"></a>
<FONT color="green">977</FONT>         * corresponding LAPACK routine DLAZQ3.&lt;/p&gt;<a name="line.977"></a>
<FONT color="green">978</FONT>         * @param start start index<a name="line.978"></a>
<FONT color="green">979</FONT>         * @param end end index<a name="line.979"></a>
<FONT color="green">980</FONT>         * @return new end (maybe deflated)<a name="line.980"></a>
<FONT color="green">981</FONT>         */<a name="line.981"></a>
<FONT color="green">982</FONT>        private int goodStep(final int start, final int end) {<a name="line.982"></a>
<FONT color="green">983</FONT>    <a name="line.983"></a>
<FONT color="green">984</FONT>            g = 0.0;<a name="line.984"></a>
<FONT color="green">985</FONT>    <a name="line.985"></a>
<FONT color="green">986</FONT>            // step 1: accepting realEigenvalues<a name="line.986"></a>
<FONT color="green">987</FONT>            int deflatedEnd = end;<a name="line.987"></a>
<FONT color="green">988</FONT>            for (boolean deflating = true; deflating;) {<a name="line.988"></a>
<FONT color="green">989</FONT>    <a name="line.989"></a>
<FONT color="green">990</FONT>                if (start &gt;= deflatedEnd) {<a name="line.990"></a>
<FONT color="green">991</FONT>                    // the array has been completely deflated<a name="line.991"></a>
<FONT color="green">992</FONT>                    return deflatedEnd;<a name="line.992"></a>
<FONT color="green">993</FONT>                }<a name="line.993"></a>
<FONT color="green">994</FONT>    <a name="line.994"></a>
<FONT color="green">995</FONT>                final int k = 4 * deflatedEnd + pingPong - 1;<a name="line.995"></a>
<FONT color="green">996</FONT>    <a name="line.996"></a>
<FONT color="green">997</FONT>                if ((start == deflatedEnd - 1) ||<a name="line.997"></a>
<FONT color="green">998</FONT>                    ((start != deflatedEnd - 2) &amp;&amp;<a name="line.998"></a>
<FONT color="green">999</FONT>                     ((work[k - 5] &lt;= TOLERANCE_2 * (sigma + work[k - 3])) ||<a name="line.999"></a>
<FONT color="green">1000</FONT>                      (work[k - 2 * pingPong - 4] &lt;= TOLERANCE_2 * work[k - 7])))) {<a name="line.1000"></a>
<FONT color="green">1001</FONT>    <a name="line.1001"></a>
<FONT color="green">1002</FONT>                    // one eigenvalue found, deflate array<a name="line.1002"></a>
<FONT color="green">1003</FONT>                    work[4 * deflatedEnd - 4] = sigma + work[4 * deflatedEnd - 4 + pingPong];<a name="line.1003"></a>
<FONT color="green">1004</FONT>                    deflatedEnd -= 1;<a name="line.1004"></a>
<FONT color="green">1005</FONT>    <a name="line.1005"></a>
<FONT color="green">1006</FONT>                } else if ((start == deflatedEnd - 2) ||<a name="line.1006"></a>
<FONT color="green">1007</FONT>                    (work[k - 9] &lt;= TOLERANCE_2 * sigma) ||<a name="line.1007"></a>
<FONT color="green">1008</FONT>                    (work[k - 2 * pingPong - 8] &lt;= TOLERANCE_2 * work[k - 11])) {<a name="line.1008"></a>
<FONT color="green">1009</FONT>    <a name="line.1009"></a>
<FONT color="green">1010</FONT>                    // two realEigenvalues found, deflate array<a name="line.1010"></a>
<FONT color="green">1011</FONT>                    if (work[k - 3] &gt; work[k - 7]) {<a name="line.1011"></a>
<FONT color="green">1012</FONT>                        final double tmp = work[k - 3];<a name="line.1012"></a>
<FONT color="green">1013</FONT>                        work[k - 3] = work[k - 7];<a name="line.1013"></a>
<FONT color="green">1014</FONT>                        work[k - 7] = tmp;<a name="line.1014"></a>
<FONT color="green">1015</FONT>                    }<a name="line.1015"></a>
<FONT color="green">1016</FONT>    <a name="line.1016"></a>
<FONT color="green">1017</FONT>                    if (work[k - 5] &gt; TOLERANCE_2 * work[k - 3]) {<a name="line.1017"></a>
<FONT color="green">1018</FONT>                        double t = 0.5 * ((work[k - 7] - work[k - 3]) + work[k - 5]);<a name="line.1018"></a>
<FONT color="green">1019</FONT>                        double s = work[k - 3] * (work[k - 5] / t);<a name="line.1019"></a>
<FONT color="green">1020</FONT>                        if (s &lt;= t) {<a name="line.1020"></a>
<FONT color="green">1021</FONT>                            s = work[k - 3] * work[k - 5] / (t * (1 + Math.sqrt(1 + s / t)));<a name="line.1021"></a>
<FONT color="green">1022</FONT>                        } else {<a name="line.1022"></a>
<FONT color="green">1023</FONT>                            s = work[k - 3] * work[k - 5] / (t + Math.sqrt(t * (t + s)));                      <a name="line.1023"></a>
<FONT color="green">1024</FONT>                        }<a name="line.1024"></a>
<FONT color="green">1025</FONT>                        t = work[k - 7] + (s + work[k - 5]);<a name="line.1025"></a>
<FONT color="green">1026</FONT>                        work[k - 3] *= work[k - 7] / t;<a name="line.1026"></a>
<FONT color="green">1027</FONT>                        work[k - 7]  = t;<a name="line.1027"></a>
<FONT color="green">1028</FONT>                    }<a name="line.1028"></a>
<FONT color="green">1029</FONT>                    work[4 * deflatedEnd - 8] = sigma + work[k - 7];<a name="line.1029"></a>
<FONT color="green">1030</FONT>                    work[4 * deflatedEnd - 4] = sigma + work[k - 3];<a name="line.1030"></a>
<FONT color="green">1031</FONT>                    deflatedEnd -= 2;<a name="line.1031"></a>
<FONT color="green">1032</FONT>                } else {<a name="line.1032"></a>
<FONT color="green">1033</FONT>    <a name="line.1033"></a>
<FONT color="green">1034</FONT>                    // no more realEigenvalues found, we need to iterate<a name="line.1034"></a>
<FONT color="green">1035</FONT>                    deflating = false;<a name="line.1035"></a>
<FONT color="green">1036</FONT>    <a name="line.1036"></a>
<FONT color="green">1037</FONT>                }<a name="line.1037"></a>
<FONT color="green">1038</FONT>    <a name="line.1038"></a>
<FONT color="green">1039</FONT>            }<a name="line.1039"></a>
<FONT color="green">1040</FONT>    <a name="line.1040"></a>
<FONT color="green">1041</FONT>            final int l = 4 * deflatedEnd + pingPong - 1;<a name="line.1041"></a>
<FONT color="green">1042</FONT>    <a name="line.1042"></a>
<FONT color="green">1043</FONT>            // step 2: flip array if needed<a name="line.1043"></a>
<FONT color="green">1044</FONT>            if ((dMin &lt;= 0) || (deflatedEnd &lt; end)) {<a name="line.1044"></a>
<FONT color="green">1045</FONT>                if (flipIfWarranted(deflatedEnd, 1)) {<a name="line.1045"></a>
<FONT color="green">1046</FONT>                    dMin2 = Math.min(dMin2, work[l - 1]);<a name="line.1046"></a>
<FONT color="green">1047</FONT>                    work[l - 1] =<a name="line.1047"></a>
<FONT color="green">1048</FONT>                        Math.min(work[l - 1],<a name="line.1048"></a>
<FONT color="green">1049</FONT>                                 Math.min(work[3 + pingPong], work[7 + pingPong]));<a name="line.1049"></a>
<FONT color="green">1050</FONT>                    work[l - 2 * pingPong] =<a name="line.1050"></a>
<FONT color="green">1051</FONT>                        Math.min(work[l - 2 * pingPong],<a name="line.1051"></a>
<FONT color="green">1052</FONT>                                 Math.min(work[6 + pingPong], work[6 + pingPong]));<a name="line.1052"></a>
<FONT color="green">1053</FONT>                    qMax  = Math.max(qMax, Math.max(work[3 + pingPong], work[7 + pingPong]));<a name="line.1053"></a>
<FONT color="green">1054</FONT>                    dMin  = -0.0;<a name="line.1054"></a>
<FONT color="green">1055</FONT>                }<a name="line.1055"></a>
<FONT color="green">1056</FONT>            }<a name="line.1056"></a>
<FONT color="green">1057</FONT>    <a name="line.1057"></a>
<FONT color="green">1058</FONT>            if ((dMin &lt; 0) ||<a name="line.1058"></a>
<FONT color="green">1059</FONT>                (MathUtils.SAFE_MIN * qMax &lt; Math.min(work[l - 1],<a name="line.1059"></a>
<FONT color="green">1060</FONT>                                                      Math.min(work[l - 9],<a name="line.1060"></a>
<FONT color="green">1061</FONT>                                                               dMin2 + work[l - 2 * pingPong])))) {<a name="line.1061"></a>
<FONT color="green">1062</FONT>                // step 3: choose a shift<a name="line.1062"></a>
<FONT color="green">1063</FONT>                computeShiftIncrement(start, deflatedEnd, end - deflatedEnd);<a name="line.1063"></a>
<FONT color="green">1064</FONT>    <a name="line.1064"></a>
<FONT color="green">1065</FONT>                // step 4a: dqds<a name="line.1065"></a>
<FONT color="green">1066</FONT>                for (boolean loop = true; loop;) {<a name="line.1066"></a>
<FONT color="green">1067</FONT>    <a name="line.1067"></a>
<FONT color="green">1068</FONT>                    // perform one dqds step with the chosen shift<a name="line.1068"></a>
<FONT color="green">1069</FONT>                    dqds(start, deflatedEnd);<a name="line.1069"></a>
<FONT color="green">1070</FONT>    <a name="line.1070"></a>
<FONT color="green">1071</FONT>                    // check result of the dqds step<a name="line.1071"></a>
<FONT color="green">1072</FONT>                    if ((dMin &gt;= 0) &amp;&amp; (dMin1 &gt; 0)) {<a name="line.1072"></a>
<FONT color="green">1073</FONT>                        // the shift was good<a name="line.1073"></a>
<FONT color="green">1074</FONT>                        updateSigma(tau);<a name="line.1074"></a>
<FONT color="green">1075</FONT>                        return deflatedEnd;<a name="line.1075"></a>
<FONT color="green">1076</FONT>                    } else if ((dMin &lt; 0.0) &amp;&amp;<a name="line.1076"></a>
<FONT color="green">1077</FONT>                               (dMin1 &gt; 0.0) &amp;&amp;<a name="line.1077"></a>
<FONT color="green">1078</FONT>                               (work[4 * deflatedEnd - 5 - pingPong] &lt; TOLERANCE * (sigma + dN1)) &amp;&amp;<a name="line.1078"></a>
<FONT color="green">1079</FONT>                               (Math.abs(dN) &lt; TOLERANCE * sigma)) {<a name="line.1079"></a>
<FONT color="green">1080</FONT>                       // convergence hidden by negative DN.<a name="line.1080"></a>
<FONT color="green">1081</FONT>                        work[4 * deflatedEnd - 3 - pingPong] = 0.0;<a name="line.1081"></a>
<FONT color="green">1082</FONT>                        dMin = 0.0;<a name="line.1082"></a>
<FONT color="green">1083</FONT>                        updateSigma(tau);<a name="line.1083"></a>
<FONT color="green">1084</FONT>                        return deflatedEnd;<a name="line.1084"></a>
<FONT color="green">1085</FONT>                    } else if (dMin &lt; 0.0) {<a name="line.1085"></a>
<FONT color="green">1086</FONT>                        // tau too big. Select new tau and try again.<a name="line.1086"></a>
<FONT color="green">1087</FONT>                        if (tType &lt; -22) {<a name="line.1087"></a>
<FONT color="green">1088</FONT>                            // failed twice. Play it safe.<a name="line.1088"></a>
<FONT color="green">1089</FONT>                            tau = 0.0;<a name="line.1089"></a>
<FONT color="green">1090</FONT>                        } else if (dMin1 &gt; 0.0) {<a name="line.1090"></a>
<FONT color="green">1091</FONT>                            // late failure. Gives excellent shift.<a name="line.1091"></a>
<FONT color="green">1092</FONT>                            tau = (tau + dMin) * (1.0 - 2.0 * MathUtils.EPSILON);<a name="line.1092"></a>
<FONT color="green">1093</FONT>                            tType -= 11;<a name="line.1093"></a>
<FONT color="green">1094</FONT>                        } else {<a name="line.1094"></a>
<FONT color="green">1095</FONT>                            // early failure. Divide by 4.<a name="line.1095"></a>
<FONT color="green">1096</FONT>                            tau *= 0.25;<a name="line.1096"></a>
<FONT color="green">1097</FONT>                            tType -= 12;<a name="line.1097"></a>
<FONT color="green">1098</FONT>                        }<a name="line.1098"></a>
<FONT color="green">1099</FONT>                    } else if (Double.isNaN(dMin)) {<a name="line.1099"></a>
<FONT color="green">1100</FONT>                        tau = 0.0;<a name="line.1100"></a>
<FONT color="green">1101</FONT>                    } else {<a name="line.1101"></a>
<FONT color="green">1102</FONT>                        // possible underflow. Play it safe.<a name="line.1102"></a>
<FONT color="green">1103</FONT>                        loop = false;<a name="line.1103"></a>
<FONT color="green">1104</FONT>                    }<a name="line.1104"></a>
<FONT color="green">1105</FONT>                }<a name="line.1105"></a>
<FONT color="green">1106</FONT>    <a name="line.1106"></a>
<FONT color="green">1107</FONT>            }<a name="line.1107"></a>
<FONT color="green">1108</FONT>    <a name="line.1108"></a>
<FONT color="green">1109</FONT>            // perform a dqd step (i.e. no shift)<a name="line.1109"></a>
<FONT color="green">1110</FONT>            dqd(start, deflatedEnd);<a name="line.1110"></a>
<FONT color="green">1111</FONT>    <a name="line.1111"></a>
<FONT color="green">1112</FONT>            return deflatedEnd;<a name="line.1112"></a>
<FONT color="green">1113</FONT>    <a name="line.1113"></a>
<FONT color="green">1114</FONT>        }<a name="line.1114"></a>
<FONT color="green">1115</FONT>    <a name="line.1115"></a>
<FONT color="green">1116</FONT>        /**<a name="line.1116"></a>
<FONT color="green">1117</FONT>         * Flip qd array if warranted.<a name="line.1117"></a>
<FONT color="green">1118</FONT>         * @param n number of rows in the block<a name="line.1118"></a>
<FONT color="green">1119</FONT>         * @param step within the array (1 for flipping all elements, 2 for flipping<a name="line.1119"></a>
<FONT color="green">1120</FONT>         * only every other element)<a name="line.1120"></a>
<FONT color="green">1121</FONT>         * @return true if qd array was flipped<a name="line.1121"></a>
<FONT color="green">1122</FONT>         */<a name="line.1122"></a>
<FONT color="green">1123</FONT>        private boolean flipIfWarranted(final int n, final int step) {<a name="line.1123"></a>
<FONT color="green">1124</FONT>            if (1.5 * work[pingPong] &lt; work[4 * (n - 1) + pingPong]) {<a name="line.1124"></a>
<FONT color="green">1125</FONT>                // flip array<a name="line.1125"></a>
<FONT color="green">1126</FONT>                for (int i = 0, j = 4 * n - 1; i &lt; j; i += 4, j -= 4) {<a name="line.1126"></a>
<FONT color="green">1127</FONT>                    for (int k = 0; k &lt; 4; k += step) {<a name="line.1127"></a>
<FONT color="green">1128</FONT>                        final double tmp = work[i + k];<a name="line.1128"></a>
<FONT color="green">1129</FONT>                        work[i + k] = work[j - k];<a name="line.1129"></a>
<FONT color="green">1130</FONT>                        work[j - k] = tmp;<a name="line.1130"></a>
<FONT color="green">1131</FONT>                    }<a name="line.1131"></a>
<FONT color="green">1132</FONT>                }<a name="line.1132"></a>
<FONT color="green">1133</FONT>                return true;<a name="line.1133"></a>
<FONT color="green">1134</FONT>            }<a name="line.1134"></a>
<FONT color="green">1135</FONT>            return false;<a name="line.1135"></a>
<FONT color="green">1136</FONT>        }<a name="line.1136"></a>
<FONT color="green">1137</FONT>    <a name="line.1137"></a>
<FONT color="green">1138</FONT>        /**<a name="line.1138"></a>
<FONT color="green">1139</FONT>         * Compute an interval containing all realEigenvalues of a block.<a name="line.1139"></a>
<FONT color="green">1140</FONT>         * @param index index of the first row of the block<a name="line.1140"></a>
<FONT color="green">1141</FONT>         * @param n number of rows of the block<a name="line.1141"></a>
<FONT color="green">1142</FONT>         * @return an interval containing the realEigenvalues<a name="line.1142"></a>
<FONT color="green">1143</FONT>         */<a name="line.1143"></a>
<FONT color="green">1144</FONT>        private double[] eigenvaluesRange(final int index, final int n) {<a name="line.1144"></a>
<FONT color="green">1145</FONT>    <a name="line.1145"></a>
<FONT color="green">1146</FONT>            // find the bounds of the spectra of the local block<a name="line.1146"></a>
<FONT color="green">1147</FONT>            final int lowerStart = 4 * main.length;<a name="line.1147"></a>
<FONT color="green">1148</FONT>            final int upperStart = 5 * main.length;<a name="line.1148"></a>
<FONT color="green">1149</FONT>            double lower = Double.POSITIVE_INFINITY;<a name="line.1149"></a>
<FONT color="green">1150</FONT>            double upper = Double.NEGATIVE_INFINITY;<a name="line.1150"></a>
<FONT color="green">1151</FONT>            for (int i = 0; i &lt; n; ++i) {<a name="line.1151"></a>
<FONT color="green">1152</FONT>                lower = Math.min(lower, work[lowerStart + index +i]);<a name="line.1152"></a>
<FONT color="green">1153</FONT>                upper = Math.max(upper, work[upperStart + index +i]);<a name="line.1153"></a>
<FONT color="green">1154</FONT>            }<a name="line.1154"></a>
<FONT color="green">1155</FONT>    <a name="line.1155"></a>
<FONT color="green">1156</FONT>            // set thresholds<a name="line.1156"></a>
<FONT color="green">1157</FONT>            final double tNorm = Math.max(Math.abs(lower), Math.abs(upper));<a name="line.1157"></a>
<FONT color="green">1158</FONT>            final double relativeTolerance = Math.sqrt(MathUtils.EPSILON);<a name="line.1158"></a>
<FONT color="green">1159</FONT>            final double absoluteTolerance = 4 * minPivot;<a name="line.1159"></a>
<FONT color="green">1160</FONT>            final int maxIter =<a name="line.1160"></a>
<FONT color="green">1161</FONT>                2 + (int) ((Math.log(tNorm + minPivot) - Math.log(minPivot)) / Math.log(2.0));<a name="line.1161"></a>
<FONT color="green">1162</FONT>            final double margin = 2 * (tNorm * MathUtils.EPSILON * n + 2 * minPivot);<a name="line.1162"></a>
<FONT color="green">1163</FONT>    <a name="line.1163"></a>
<FONT color="green">1164</FONT>            // search lower eigenvalue<a name="line.1164"></a>
<FONT color="green">1165</FONT>            double left  = lower - margin;<a name="line.1165"></a>
<FONT color="green">1166</FONT>            double right = upper + margin;<a name="line.1166"></a>
<FONT color="green">1167</FONT>            for (int i = 0; i &lt; maxIter; ++i) {<a name="line.1167"></a>
<FONT color="green">1168</FONT>    <a name="line.1168"></a>
<FONT color="green">1169</FONT>                final double range = right - left;<a name="line.1169"></a>
<FONT color="green">1170</FONT>                if ((range &lt; absoluteTolerance) ||<a name="line.1170"></a>
<FONT color="green">1171</FONT>                    (range &lt; relativeTolerance * Math.max(Math.abs(left), Math.abs(right)))) {<a name="line.1171"></a>
<FONT color="green">1172</FONT>                    // search has converged<a name="line.1172"></a>
<FONT color="green">1173</FONT>                    break;<a name="line.1173"></a>
<FONT color="green">1174</FONT>                }<a name="line.1174"></a>
<FONT color="green">1175</FONT>    <a name="line.1175"></a>
<FONT color="green">1176</FONT>                final double middle = 0.5 * (left + right);<a name="line.1176"></a>
<FONT color="green">1177</FONT>                if (countEigenValues(middle, index, n) &gt;= 1) {<a name="line.1177"></a>
<FONT color="green">1178</FONT>                    right = middle;<a name="line.1178"></a>
<FONT color="green">1179</FONT>                } else {<a name="line.1179"></a>
<FONT color="green">1180</FONT>                    left = middle;<a name="line.1180"></a>
<FONT color="green">1181</FONT>                }<a name="line.1181"></a>
<FONT color="green">1182</FONT>    <a name="line.1182"></a>
<FONT color="green">1183</FONT>            }<a name="line.1183"></a>
<FONT color="green">1184</FONT>            lower = Math.max(lower, left - 100 * MathUtils.EPSILON * Math.abs(left));<a name="line.1184"></a>
<FONT color="green">1185</FONT>    <a name="line.1185"></a>
<FONT color="green">1186</FONT>            // search upper eigenvalue<a name="line.1186"></a>
<FONT color="green">1187</FONT>            left  = lower - margin;<a name="line.1187"></a>
<FONT color="green">1188</FONT>            right = upper + margin;<a name="line.1188"></a>
<FONT color="green">1189</FONT>            for (int i = 0; i &lt; maxIter; ++i) {<a name="line.1189"></a>
<FONT color="green">1190</FONT>    <a name="line.1190"></a>
<FONT color="green">1191</FONT>                final double range = right - left;<a name="line.1191"></a>
<FONT color="green">1192</FONT>                if ((range &lt; absoluteTolerance) ||<a name="line.1192"></a>
<FONT color="green">1193</FONT>                    (range &lt; relativeTolerance * Math.max(Math.abs(left), Math.abs(right)))) {<a name="line.1193"></a>
<FONT color="green">1194</FONT>                    // search has converged<a name="line.1194"></a>
<FONT color="green">1195</FONT>                    break;<a name="line.1195"></a>
<FONT color="green">1196</FONT>                }<a name="line.1196"></a>
<FONT color="green">1197</FONT>    <a name="line.1197"></a>
<FONT color="green">1198</FONT>                final double middle = 0.5 * (left + right);<a name="line.1198"></a>
<FONT color="green">1199</FONT>                if (countEigenValues(middle, index, n) &gt;= n) {<a name="line.1199"></a>
<FONT color="green">1200</FONT>                    right = middle;<a name="line.1200"></a>
<FONT color="green">1201</FONT>                } else {<a name="line.1201"></a>
<FONT color="green">1202</FONT>                    left = middle;<a name="line.1202"></a>
<FONT color="green">1203</FONT>                }<a name="line.1203"></a>
<FONT color="green">1204</FONT>    <a name="line.1204"></a>
<FONT color="green">1205</FONT>            }<a name="line.1205"></a>
<FONT color="green">1206</FONT>            upper = Math.min(upper, right + 100 * MathUtils.EPSILON * Math.abs(right));<a name="line.1206"></a>
<FONT color="green">1207</FONT>    <a name="line.1207"></a>
<FONT color="green">1208</FONT>            return new double[] { lower, upper };<a name="line.1208"></a>
<FONT color="green">1209</FONT>    <a name="line.1209"></a>
<FONT color="green">1210</FONT>        }<a name="line.1210"></a>
<FONT color="green">1211</FONT>    <a name="line.1211"></a>
<FONT color="green">1212</FONT>        /**<a name="line.1212"></a>
<FONT color="green">1213</FONT>         * Count the number of realEigenvalues below a point.<a name="line.1213"></a>
<FONT color="green">1214</FONT>         * @param t value below which we must count the number of realEigenvalues<a name="line.1214"></a>
<FONT color="green">1215</FONT>         * @param index index of the first row of the block<a name="line.1215"></a>
<FONT color="green">1216</FONT>         * @param n number of rows of the block<a name="line.1216"></a>
<FONT color="green">1217</FONT>         * @return number of realEigenvalues smaller than t<a name="line.1217"></a>
<FONT color="green">1218</FONT>         */<a name="line.1218"></a>
<FONT color="green">1219</FONT>        private int countEigenValues(final double t, final int index, final int n) {<a name="line.1219"></a>
<FONT color="green">1220</FONT>            double ratio = main[index] - t;<a name="line.1220"></a>
<FONT color="green">1221</FONT>            int count = (ratio &gt; 0) ? 0 : 1;<a name="line.1221"></a>
<FONT color="green">1222</FONT>            for (int i = 1; i &lt; n; ++i) {<a name="line.1222"></a>
<FONT color="green">1223</FONT>                ratio = main[index + i] - squaredSecondary[index + i - 1] / ratio - t;<a name="line.1223"></a>
<FONT color="green">1224</FONT>                if (ratio &lt;= 0) {<a name="line.1224"></a>
<FONT color="green">1225</FONT>                    ++count;<a name="line.1225"></a>
<FONT color="green">1226</FONT>                }<a name="line.1226"></a>
<FONT color="green">1227</FONT>            }<a name="line.1227"></a>
<FONT color="green">1228</FONT>            return count;<a name="line.1228"></a>
<FONT color="green">1229</FONT>        }<a name="line.1229"></a>
<FONT color="green">1230</FONT>    <a name="line.1230"></a>
<FONT color="green">1231</FONT>        /**<a name="line.1231"></a>
<FONT color="green">1232</FONT>         * Decompose the shifted tridiagonal matrix T-&amp;lambda;I as LDL&lt;sup&gt;T&lt;/sup&gt;.<a name="line.1232"></a>
<FONT color="green">1233</FONT>         * &lt;p&gt;A shifted symmetric tridiagonal matrix T can be decomposed as<a name="line.1233"></a>
<FONT color="green">1234</FONT>         * LDL&lt;sup&gt;T&lt;/sup&gt; where L is a lower bidiagonal matrix with unit diagonal<a name="line.1234"></a>
<FONT color="green">1235</FONT>         * and D is a diagonal matrix. This method is an implementation of<a name="line.1235"></a>
<FONT color="green">1236</FONT>         * algorithm 4.4.7 from Dhillon's thesis.&lt;/p&gt;<a name="line.1236"></a>
<FONT color="green">1237</FONT>         * @param lambda shift to add to the matrix before decomposing it<a name="line.1237"></a>
<FONT color="green">1238</FONT>         * to ensure it is positive definite<a name="line.1238"></a>
<FONT color="green">1239</FONT>         * @param index index of the first row of the block<a name="line.1239"></a>
<FONT color="green">1240</FONT>         * @param n number of rows of the block<a name="line.1240"></a>
<FONT color="green">1241</FONT>         */<a name="line.1241"></a>
<FONT color="green">1242</FONT>        private void ldlTDecomposition(final double lambda, final int index, final int n) {<a name="line.1242"></a>
<FONT color="green">1243</FONT>            double di = main[index] - lambda;<a name="line.1243"></a>
<FONT color="green">1244</FONT>            work[0] = Math.abs(di);<a name="line.1244"></a>
<FONT color="green">1245</FONT>            for (int i = 1; i &lt; n; ++i) {<a name="line.1245"></a>
<FONT color="green">1246</FONT>                final int    fourI = 4 * i;<a name="line.1246"></a>
<FONT color="green">1247</FONT>                final double eiM1  = secondary[index + i - 1];<a name="line.1247"></a>
<FONT color="green">1248</FONT>                final double ratio = eiM1 / di;<a name="line.1248"></a>
<FONT color="green">1249</FONT>                work[fourI - 2] = ratio * ratio * Math.abs(di);<a name="line.1249"></a>
<FONT color="green">1250</FONT>                di = (main[index + i] - lambda) - eiM1 * ratio;<a name="line.1250"></a>
<FONT color="green">1251</FONT>                work[fourI] = Math.abs(di);<a name="line.1251"></a>
<FONT color="green">1252</FONT>            }<a name="line.1252"></a>
<FONT color="green">1253</FONT>        }<a name="line.1253"></a>
<FONT color="green">1254</FONT>    <a name="line.1254"></a>
<FONT color="green">1255</FONT>        /**<a name="line.1255"></a>
<FONT color="green">1256</FONT>         * Perform a dqds step, using current shift increment.<a name="line.1256"></a>
<FONT color="green">1257</FONT>         * &lt;p&gt;This implementation is a translation of the LAPACK routine DLASQ5.&lt;/p&gt;<a name="line.1257"></a>
<FONT color="green">1258</FONT>         * @param start start index<a name="line.1258"></a>
<FONT color="green">1259</FONT>         * @param end end index<a name="line.1259"></a>
<FONT color="green">1260</FONT>         */<a name="line.1260"></a>
<FONT color="green">1261</FONT>        private void dqds(final int start, final int end) {<a name="line.1261"></a>
<FONT color="green">1262</FONT>    <a name="line.1262"></a>
<FONT color="green">1263</FONT>            eMin = work[4 * start + pingPong + 4];<a name="line.1263"></a>
<FONT color="green">1264</FONT>            double d = work[4 * start + pingPong] - tau;<a name="line.1264"></a>
<FONT color="green">1265</FONT>            dMin = d;<a name="line.1265"></a>
<FONT color="green">1266</FONT>            dMin1 = -work[4 * start + pingPong];<a name="line.1266"></a>
<FONT color="green">1267</FONT>    <a name="line.1267"></a>
<FONT color="green">1268</FONT>            if (pingPong == 0) {<a name="line.1268"></a>
<FONT color="green">1269</FONT>                for (int j4 = 4 * start + 3; j4 &lt;= 4 * (end - 3); j4 += 4) {<a name="line.1269"></a>
<FONT color="green">1270</FONT>                    work[j4 - 2] = d + work[j4 - 1];<a name="line.1270"></a>
<FONT color="green">1271</FONT>                    final double tmp = work[j4 + 1] / work[j4 - 2];<a name="line.1271"></a>
<FONT color="green">1272</FONT>                    d = d * tmp - tau;<a name="line.1272"></a>
<FONT color="green">1273</FONT>                    dMin = Math.min(dMin, d);<a name="line.1273"></a>
<FONT color="green">1274</FONT>                    work[j4] = work[j4 - 1] * tmp;<a name="line.1274"></a>
<FONT color="green">1275</FONT>                    eMin = Math.min(work[j4], eMin);<a name="line.1275"></a>
<FONT color="green">1276</FONT>                }<a name="line.1276"></a>
<FONT color="green">1277</FONT>            } else {<a name="line.1277"></a>
<FONT color="green">1278</FONT>                for (int j4 = 4 * start + 3; j4 &lt;= 4 * (end - 3); j4 += 4) {<a name="line.1278"></a>
<FONT color="green">1279</FONT>                    work[j4 - 3] = d + work[j4];<a name="line.1279"></a>
<FONT color="green">1280</FONT>                    final double tmp = work[j4 + 2] / work[j4 - 3];<a name="line.1280"></a>
<FONT color="green">1281</FONT>                    d = d * tmp - tau;<a name="line.1281"></a>
<FONT color="green">1282</FONT>                    dMin = Math.min(dMin, d);<a name="line.1282"></a>
<FONT color="green">1283</FONT>                    work[j4 - 1] = work[j4] * tmp;<a name="line.1283"></a>
<FONT color="green">1284</FONT>                    eMin = Math.min(work[j4 - 1], eMin);<a name="line.1284"></a>
<FONT color="green">1285</FONT>                }<a name="line.1285"></a>
<FONT color="green">1286</FONT>            }<a name="line.1286"></a>
<FONT color="green">1287</FONT>    <a name="line.1287"></a>
<FONT color="green">1288</FONT>            // unroll last two steps.<a name="line.1288"></a>
<FONT color="green">1289</FONT>            dN2 = d;<a name="line.1289"></a>
<FONT color="green">1290</FONT>            dMin2 = dMin;<a name="line.1290"></a>
<FONT color="green">1291</FONT>            int j4 = 4 * (end - 2) - pingPong - 1;<a name="line.1291"></a>
<FONT color="green">1292</FONT>            int j4p2 = j4 + 2 * pingPong - 1;<a name="line.1292"></a>
<FONT color="green">1293</FONT>            work[j4 - 2] = dN2 + work[j4p2];<a name="line.1293"></a>
<FONT color="green">1294</FONT>            work[j4] = work[j4p2 + 2] * (work[j4p2] / work[j4 - 2]);<a name="line.1294"></a>
<FONT color="green">1295</FONT>            dN1 = work[j4p2 + 2] * (dN2 / work[j4 - 2]) - tau;<a name="line.1295"></a>
<FONT color="green">1296</FONT>            dMin = Math.min(dMin, dN1);<a name="line.1296"></a>
<FONT color="green">1297</FONT>    <a name="line.1297"></a>
<FONT color="green">1298</FONT>            dMin1 = dMin;<a name="line.1298"></a>
<FONT color="green">1299</FONT>            j4 = j4 + 4;<a name="line.1299"></a>
<FONT color="green">1300</FONT>            j4p2 = j4 + 2 * pingPong - 1;<a name="line.1300"></a>
<FONT color="green">1301</FONT>            work[j4 - 2] = dN1 + work[j4p2];<a name="line.1301"></a>
<FONT color="green">1302</FONT>            work[j4] = work[j4p2 + 2] * (work[j4p2] / work[j4 - 2]);<a name="line.1302"></a>
<FONT color="green">1303</FONT>            dN = work[j4p2 + 2] * (dN1 / work[j4 - 2]) - tau;<a name="line.1303"></a>
<FONT color="green">1304</FONT>            dMin = Math.min(dMin, dN);<a name="line.1304"></a>
<FONT color="green">1305</FONT>    <a name="line.1305"></a>
<FONT color="green">1306</FONT>            work[j4 + 2] = dN;<a name="line.1306"></a>
<FONT color="green">1307</FONT>            work[4 * end - pingPong - 1] = eMin;<a name="line.1307"></a>
<FONT color="green">1308</FONT>    <a name="line.1308"></a>
<FONT color="green">1309</FONT>        }<a name="line.1309"></a>
<FONT color="green">1310</FONT>    <a name="line.1310"></a>
<FONT color="green">1311</FONT>    <a name="line.1311"></a>
<FONT color="green">1312</FONT>        /**<a name="line.1312"></a>
<FONT color="green">1313</FONT>         * Perform a dqd step.<a name="line.1313"></a>
<FONT color="green">1314</FONT>         * &lt;p&gt;This implementation is a translation of the LAPACK routine DLASQ6.&lt;/p&gt;<a name="line.1314"></a>
<FONT color="green">1315</FONT>         * @param start start index<a name="line.1315"></a>
<FONT color="green">1316</FONT>         * @param end end index<a name="line.1316"></a>
<FONT color="green">1317</FONT>         */<a name="line.1317"></a>
<FONT color="green">1318</FONT>        private void dqd(final int start, final int end) {<a name="line.1318"></a>
<FONT color="green">1319</FONT>    <a name="line.1319"></a>
<FONT color="green">1320</FONT>            eMin = work[4 * start + pingPong + 4];<a name="line.1320"></a>
<FONT color="green">1321</FONT>            double d = work[4 * start + pingPong];<a name="line.1321"></a>
<FONT color="green">1322</FONT>            dMin = d;<a name="line.1322"></a>
<FONT color="green">1323</FONT>    <a name="line.1323"></a>
<FONT color="green">1324</FONT>            if (pingPong == 0) {<a name="line.1324"></a>
<FONT color="green">1325</FONT>                for (int j4 = 4 * start + 3; j4 &lt; 4 * (end - 3); j4 += 4) {<a name="line.1325"></a>
<FONT color="green">1326</FONT>                    work[j4 - 2] = d + work[j4 - 1];<a name="line.1326"></a>
<FONT color="green">1327</FONT>                    if (work[j4 - 2] == 0.0) {<a name="line.1327"></a>
<FONT color="green">1328</FONT>                        work[j4] = 0.0;<a name="line.1328"></a>
<FONT color="green">1329</FONT>                        d = work[j4 + 1];<a name="line.1329"></a>
<FONT color="green">1330</FONT>                        dMin = d;<a name="line.1330"></a>
<FONT color="green">1331</FONT>                        eMin = 0.0;<a name="line.1331"></a>
<FONT color="green">1332</FONT>                    } else if ((MathUtils.SAFE_MIN * work[j4 + 1] &lt; work[j4 - 2]) &amp;&amp;<a name="line.1332"></a>
<FONT color="green">1333</FONT>                               (MathUtils.SAFE_MIN * work[j4 - 2] &lt; work[j4 + 1])) {<a name="line.1333"></a>
<FONT color="green">1334</FONT>                        final double tmp = work[j4 + 1] / work[j4 - 2];<a name="line.1334"></a>
<FONT color="green">1335</FONT>                        work[j4] = work[j4 - 1] * tmp;<a name="line.1335"></a>
<FONT color="green">1336</FONT>                        d *= tmp;<a name="line.1336"></a>
<FONT color="green">1337</FONT>                    } else {<a name="line.1337"></a>
<FONT color="green">1338</FONT>                        work[j4] = work[j4 + 1] * (work[j4 - 1] / work[j4 - 2]);<a name="line.1338"></a>
<FONT color="green">1339</FONT>                        d *= work[j4 + 1] / work[j4 - 2];<a name="line.1339"></a>
<FONT color="green">1340</FONT>                    }<a name="line.1340"></a>
<FONT color="green">1341</FONT>                    dMin = Math.min(dMin, d);<a name="line.1341"></a>
<FONT color="green">1342</FONT>                    eMin = Math.min(eMin, work[j4]);<a name="line.1342"></a>
<FONT color="green">1343</FONT>                }<a name="line.1343"></a>
<FONT color="green">1344</FONT>            } else {<a name="line.1344"></a>
<FONT color="green">1345</FONT>                for (int j4 = 4 * start + 3; j4 &lt; 4 * (end - 3); j4 += 4) {<a name="line.1345"></a>
<FONT color="green">1346</FONT>                    work[j4 - 3] = d + work[j4];<a name="line.1346"></a>
<FONT color="green">1347</FONT>                    if (work[j4 - 3] == 0.0) {<a name="line.1347"></a>
<FONT color="green">1348</FONT>                        work[j4 - 1] = 0.0;<a name="line.1348"></a>
<FONT color="green">1349</FONT>                        d = work[j4 + 2];<a name="line.1349"></a>
<FONT color="green">1350</FONT>                        dMin = d;<a name="line.1350"></a>
<FONT color="green">1351</FONT>                        eMin = 0.0;<a name="line.1351"></a>
<FONT color="green">1352</FONT>                    } else if ((MathUtils.SAFE_MIN * work[j4 + 2] &lt; work[j4 - 3]) &amp;&amp;<a name="line.1352"></a>
<FONT color="green">1353</FONT>                               (MathUtils.SAFE_MIN * work[j4 - 3] &lt; work[j4 + 2])) {<a name="line.1353"></a>
<FONT color="green">1354</FONT>                        final double tmp = work[j4 + 2] / work[j4 - 3];<a name="line.1354"></a>
<FONT color="green">1355</FONT>                        work[j4 - 1] = work[j4] * tmp;<a name="line.1355"></a>
<FONT color="green">1356</FONT>                        d *= tmp;<a name="line.1356"></a>
<FONT color="green">1357</FONT>                    } else {<a name="line.1357"></a>
<FONT color="green">1358</FONT>                        work[j4 - 1] = work[j4 + 2] * (work[j4] / work[j4 - 3]);<a name="line.1358"></a>
<FONT color="green">1359</FONT>                        d *= work[j4 + 2] / work[j4 - 3];<a name="line.1359"></a>
<FONT color="green">1360</FONT>                    }<a name="line.1360"></a>
<FONT color="green">1361</FONT>                    dMin = Math.min(dMin, d);<a name="line.1361"></a>
<FONT color="green">1362</FONT>                    eMin = Math.min(eMin, work[j4 - 1]);<a name="line.1362"></a>
<FONT color="green">1363</FONT>                }<a name="line.1363"></a>
<FONT color="green">1364</FONT>            }<a name="line.1364"></a>
<FONT color="green">1365</FONT>    <a name="line.1365"></a>
<FONT color="green">1366</FONT>            // Unroll last two steps<a name="line.1366"></a>
<FONT color="green">1367</FONT>            dN2   = d;<a name="line.1367"></a>
<FONT color="green">1368</FONT>            dMin2 = dMin;<a name="line.1368"></a>
<FONT color="green">1369</FONT>            int j4 = 4 * (end - 2) - pingPong - 1;<a name="line.1369"></a>
<FONT color="green">1370</FONT>            int j4p2 = j4 + 2 * pingPong - 1;<a name="line.1370"></a>
<FONT color="green">1371</FONT>            work[j4 - 2] = dN2 + work[j4p2];<a name="line.1371"></a>
<FONT color="green">1372</FONT>            if (work[j4 - 2] == 0.0) {<a name="line.1372"></a>
<FONT color="green">1373</FONT>                work[j4] = 0.0;<a name="line.1373"></a>
<FONT color="green">1374</FONT>                dN1  = work[j4p2 + 2];<a name="line.1374"></a>
<FONT color="green">1375</FONT>                dMin = dN1;<a name="line.1375"></a>
<FONT color="green">1376</FONT>                eMin = 0.0;<a name="line.1376"></a>
<FONT color="green">1377</FONT>            } else if ((MathUtils.SAFE_MIN * work[j4p2 + 2] &lt; work[j4 - 2]) &amp;&amp;<a name="line.1377"></a>
<FONT color="green">1378</FONT>                       (MathUtils.SAFE_MIN * work[j4 - 2] &lt; work[j4p2 + 2])) {<a name="line.1378"></a>
<FONT color="green">1379</FONT>                final double tmp = work[j4p2 + 2] / work[j4 - 2];<a name="line.1379"></a>
<FONT color="green">1380</FONT>                work[j4] = work[j4p2] * tmp;<a name="line.1380"></a>
<FONT color="green">1381</FONT>                dN1 = dN2 * tmp;<a name="line.1381"></a>
<FONT color="green">1382</FONT>            } else {<a name="line.1382"></a>
<FONT color="green">1383</FONT>                work[j4] = work[j4p2 + 2] * (work[j4p2] / work[j4 - 2]);<a name="line.1383"></a>
<FONT color="green">1384</FONT>                dN1 = work[j4p2 + 2] * (dN2 / work[j4 - 2]);<a name="line.1384"></a>
<FONT color="green">1385</FONT>            }<a name="line.1385"></a>
<FONT color="green">1386</FONT>            dMin = Math.min(dMin, dN1);<a name="line.1386"></a>
<FONT color="green">1387</FONT>    <a name="line.1387"></a>
<FONT color="green">1388</FONT>            dMin1 = dMin;<a name="line.1388"></a>
<FONT color="green">1389</FONT>            j4 = j4 + 4;<a name="line.1389"></a>
<FONT color="green">1390</FONT>            j4p2 = j4 + 2 * pingPong - 1;<a name="line.1390"></a>
<FONT color="green">1391</FONT>            work[j4 - 2] = dN1 + work[j4p2];<a name="line.1391"></a>
<FONT color="green">1392</FONT>            if (work[j4 - 2] == 0.0) {<a name="line.1392"></a>
<FONT color="green">1393</FONT>                work[j4] = 0.0;<a name="line.1393"></a>
<FONT color="green">1394</FONT>                dN   = work[j4p2 + 2];<a name="line.1394"></a>
<FONT color="green">1395</FONT>                dMin = dN;<a name="line.1395"></a>
<FONT color="green">1396</FONT>                eMin = 0.0;<a name="line.1396"></a>
<FONT color="green">1397</FONT>            } else if ((MathUtils.SAFE_MIN * work[j4p2 + 2] &lt; work[j4 - 2]) &amp;&amp;<a name="line.1397"></a>
<FONT color="green">1398</FONT>                       (MathUtils.SAFE_MIN * work[j4 - 2] &lt; work[j4p2 + 2])) {<a name="line.1398"></a>
<FONT color="green">1399</FONT>                final double tmp = work[j4p2 + 2] / work[j4 - 2];<a name="line.1399"></a>
<FONT color="green">1400</FONT>                work[j4] = work[j4p2] * tmp;<a name="line.1400"></a>
<FONT color="green">1401</FONT>                dN = dN1 * tmp;<a name="line.1401"></a>
<FONT color="green">1402</FONT>            } else {<a name="line.1402"></a>
<FONT color="green">1403</FONT>                work[j4] = work[j4p2 + 2] * (work[j4p2] / work[j4 - 2]);<a name="line.1403"></a>
<FONT color="green">1404</FONT>                dN = work[j4p2 + 2] * (dN1 / work[j4 - 2]);<a name="line.1404"></a>
<FONT color="green">1405</FONT>            }<a name="line.1405"></a>
<FONT color="green">1406</FONT>            dMin = Math.min(dMin, dN);<a name="line.1406"></a>
<FONT color="green">1407</FONT>    <a name="line.1407"></a>
<FONT color="green">1408</FONT>            work[j4 + 2] = dN;<a name="line.1408"></a>
<FONT color="green">1409</FONT>            work[4 * end - pingPong - 1] = eMin;<a name="line.1409"></a>
<FONT color="green">1410</FONT>    <a name="line.1410"></a>
<FONT color="green">1411</FONT>        }<a name="line.1411"></a>
<FONT color="green">1412</FONT>    <a name="line.1412"></a>
<FONT color="green">1413</FONT>        /**<a name="line.1413"></a>
<FONT color="green">1414</FONT>         * Compute the shift increment as an estimate of the smallest eigenvalue.<a name="line.1414"></a>
<FONT color="green">1415</FONT>         * &lt;p&gt;This implementation is a translation of the LAPACK routine DLAZQ4.&lt;/p&gt;<a name="line.1415"></a>
<FONT color="green">1416</FONT>         * @param start start index<a name="line.1416"></a>
<FONT color="green">1417</FONT>         * @param end end index<a name="line.1417"></a>
<FONT color="green">1418</FONT>         * @param deflated number of realEigenvalues just deflated<a name="line.1418"></a>
<FONT color="green">1419</FONT>         */<a name="line.1419"></a>
<FONT color="green">1420</FONT>        private void computeShiftIncrement(final int start, final int end, final int deflated) {<a name="line.1420"></a>
<FONT color="green">1421</FONT>    <a name="line.1421"></a>
<FONT color="green">1422</FONT>            final double cnst1 = 0.563;<a name="line.1422"></a>
<FONT color="green">1423</FONT>            final double cnst2 = 1.010;<a name="line.1423"></a>
<FONT color="green">1424</FONT>            final double cnst3 = 1.05;<a name="line.1424"></a>
<FONT color="green">1425</FONT>    <a name="line.1425"></a>
<FONT color="green">1426</FONT>            // a negative dMin forces the shift to take that absolute value<a name="line.1426"></a>
<FONT color="green">1427</FONT>            // tType records the type of shift.<a name="line.1427"></a>
<FONT color="green">1428</FONT>            if (dMin &lt;= 0.0) {<a name="line.1428"></a>
<FONT color="green">1429</FONT>                tau = -dMin;<a name="line.1429"></a>
<FONT color="green">1430</FONT>                tType = -1;<a name="line.1430"></a>
<FONT color="green">1431</FONT>                return;<a name="line.1431"></a>
<FONT color="green">1432</FONT>            }<a name="line.1432"></a>
<FONT color="green">1433</FONT>    <a name="line.1433"></a>
<FONT color="green">1434</FONT>            int nn = 4 * end + pingPong - 1;<a name="line.1434"></a>
<FONT color="green">1435</FONT>            switch (deflated) {<a name="line.1435"></a>
<FONT color="green">1436</FONT>    <a name="line.1436"></a>
<FONT color="green">1437</FONT>            case 0 : // no realEigenvalues deflated. <a name="line.1437"></a>
<FONT color="green">1438</FONT>                if (dMin == dN || dMin == dN1) {<a name="line.1438"></a>
<FONT color="green">1439</FONT>    <a name="line.1439"></a>
<FONT color="green">1440</FONT>                    double b1 = Math.sqrt(work[nn - 3]) * Math.sqrt(work[nn - 5]);<a name="line.1440"></a>
<FONT color="green">1441</FONT>                    double b2 = Math.sqrt(work[nn - 7]) * Math.sqrt(work[nn - 9]);<a name="line.1441"></a>
<FONT color="green">1442</FONT>                    double a2 = work[nn - 7] + work[nn - 5];<a name="line.1442"></a>
<FONT color="green">1443</FONT>    <a name="line.1443"></a>
<FONT color="green">1444</FONT>                    if (dMin == dN &amp;&amp; dMin1 == dN1) {<a name="line.1444"></a>
<FONT color="green">1445</FONT>                        // cases 2 and 3. <a name="line.1445"></a>
<FONT color="green">1446</FONT>                        final double gap2 = dMin2 - a2 - dMin2 * 0.25;<a name="line.1446"></a>
<FONT color="green">1447</FONT>                        final double gap1 = a2 - dN - ((gap2 &gt; 0.0 &amp;&amp; gap2 &gt; b2) ? (b2 / gap2) * b2 : (b1 + b2));<a name="line.1447"></a>
<FONT color="green">1448</FONT>                        if (gap1 &gt; 0.0 &amp;&amp; gap1 &gt; b1) {<a name="line.1448"></a>
<FONT color="green">1449</FONT>                            tau   = Math.max(dN - (b1 / gap1) * b1, 0.5 * dMin);<a name="line.1449"></a>
<FONT color="green">1450</FONT>                            tType = -2;<a name="line.1450"></a>
<FONT color="green">1451</FONT>                        } else {<a name="line.1451"></a>
<FONT color="green">1452</FONT>                            double s = 0.0;<a name="line.1452"></a>
<FONT color="green">1453</FONT>                            if (dN &gt; b1) {<a name="line.1453"></a>
<FONT color="green">1454</FONT>                                s = dN - b1;<a name="line.1454"></a>
<FONT color="green">1455</FONT>                            }<a name="line.1455"></a>
<FONT color="green">1456</FONT>                            if (a2 &gt; (b1 + b2)) {<a name="line.1456"></a>
<FONT color="green">1457</FONT>                                s = Math.min(s, a2 - (b1 + b2));<a name="line.1457"></a>
<FONT color="green">1458</FONT>                            }<a name="line.1458"></a>
<FONT color="green">1459</FONT>                            tau   = Math.max(s, 0.333 * dMin);<a name="line.1459"></a>
<FONT color="green">1460</FONT>                            tType = -3;<a name="line.1460"></a>
<FONT color="green">1461</FONT>                        }<a name="line.1461"></a>
<FONT color="green">1462</FONT>                    } else {<a name="line.1462"></a>
<FONT color="green">1463</FONT>                        // case 4.<a name="line.1463"></a>
<FONT color="green">1464</FONT>                        tType = -4;<a name="line.1464"></a>
<FONT color="green">1465</FONT>                        double s = 0.25 * dMin;<a name="line.1465"></a>
<FONT color="green">1466</FONT>                        double gam;<a name="line.1466"></a>
<FONT color="green">1467</FONT>                        int np;<a name="line.1467"></a>
<FONT color="green">1468</FONT>                        if (dMin == dN) {<a name="line.1468"></a>
<FONT color="green">1469</FONT>                            gam = dN;<a name="line.1469"></a>
<FONT color="green">1470</FONT>                            a2 = 0.0;<a name="line.1470"></a>
<FONT color="green">1471</FONT>                            if (work[nn - 5]  &gt;  work[nn - 7]) {<a name="line.1471"></a>
<FONT color="green">1472</FONT>                                return;<a name="line.1472"></a>
<FONT color="green">1473</FONT>                            }<a name="line.1473"></a>
<FONT color="green">1474</FONT>                            b2 = work[nn - 5] / work[nn - 7];<a name="line.1474"></a>
<FONT color="green">1475</FONT>                            np = nn - 9;<a name="line.1475"></a>
<FONT color="green">1476</FONT>                        } else {<a name="line.1476"></a>
<FONT color="green">1477</FONT>                            np = nn - 2 * pingPong;<a name="line.1477"></a>
<FONT color="green">1478</FONT>                            b2 = work[np - 2];<a name="line.1478"></a>
<FONT color="green">1479</FONT>                            gam = dN1;<a name="line.1479"></a>
<FONT color="green">1480</FONT>                            if (work[np - 4]  &gt;  work[np - 2]) {<a name="line.1480"></a>
<FONT color="green">1481</FONT>                                return;<a name="line.1481"></a>
<FONT color="green">1482</FONT>                            }<a name="line.1482"></a>
<FONT color="green">1483</FONT>                            a2 = work[np - 4] / work[np - 2];<a name="line.1483"></a>
<FONT color="green">1484</FONT>                            if (work[nn - 9]  &gt;  work[nn - 11]) {<a name="line.1484"></a>
<FONT color="green">1485</FONT>                                return;<a name="line.1485"></a>
<FONT color="green">1486</FONT>                            }<a name="line.1486"></a>
<FONT color="green">1487</FONT>                            b2 = work[nn - 9] / work[nn - 11];<a name="line.1487"></a>
<FONT color="green">1488</FONT>                            np = nn - 13;<a name="line.1488"></a>
<FONT color="green">1489</FONT>                        }<a name="line.1489"></a>
<FONT color="green">1490</FONT>    <a name="line.1490"></a>
<FONT color="green">1491</FONT>                        // approximate contribution to norm squared from i &lt; nn-1.<a name="line.1491"></a>
<FONT color="green">1492</FONT>                        a2 = a2 + b2;<a name="line.1492"></a>
<FONT color="green">1493</FONT>                        for (int i4 = np; i4 &gt;= 4 * start + 2 + pingPong; i4 -= 4) {<a name="line.1493"></a>
<FONT color="green">1494</FONT>                            if(b2 == 0.0) {<a name="line.1494"></a>
<FONT color="green">1495</FONT>                                break;<a name="line.1495"></a>
<FONT color="green">1496</FONT>                            }<a name="line.1496"></a>
<FONT color="green">1497</FONT>                            b1 = b2;<a name="line.1497"></a>
<FONT color="green">1498</FONT>                            if (work[i4]  &gt;  work[i4 - 2]) {<a name="line.1498"></a>
<FONT color="green">1499</FONT>                                return;<a name="line.1499"></a>
<FONT color="green">1500</FONT>                            }<a name="line.1500"></a>
<FONT color="green">1501</FONT>                            b2 = b2 * (work[i4] / work[i4 - 2]);<a name="line.1501"></a>
<FONT color="green">1502</FONT>                            a2 = a2 + b2;<a name="line.1502"></a>
<FONT color="green">1503</FONT>                            if (100 * Math.max(b2, b1) &lt; a2 || cnst1 &lt; a2) {<a name="line.1503"></a>
<FONT color="green">1504</FONT>                                break;<a name="line.1504"></a>
<FONT color="green">1505</FONT>                            }<a name="line.1505"></a>
<FONT color="green">1506</FONT>                        }<a name="line.1506"></a>
<FONT color="green">1507</FONT>                        a2 = cnst3 * a2;<a name="line.1507"></a>
<FONT color="green">1508</FONT>    <a name="line.1508"></a>
<FONT color="green">1509</FONT>                        // rayleigh quotient residual bound.<a name="line.1509"></a>
<FONT color="green">1510</FONT>                        if (a2 &lt; cnst1) {<a name="line.1510"></a>
<FONT color="green">1511</FONT>                            s = gam * (1 - Math.sqrt(a2)) / (1 + a2);<a name="line.1511"></a>
<FONT color="green">1512</FONT>                        }<a name="line.1512"></a>
<FONT color="green">1513</FONT>                        tau = s;<a name="line.1513"></a>
<FONT color="green">1514</FONT>    <a name="line.1514"></a>
<FONT color="green">1515</FONT>                    }<a name="line.1515"></a>
<FONT color="green">1516</FONT>                } else if (dMin == dN2) {<a name="line.1516"></a>
<FONT color="green">1517</FONT>    <a name="line.1517"></a>
<FONT color="green">1518</FONT>                    // case 5.<a name="line.1518"></a>
<FONT color="green">1519</FONT>                    tType = -5;<a name="line.1519"></a>
<FONT color="green">1520</FONT>                    double s = 0.25 * dMin;<a name="line.1520"></a>
<FONT color="green">1521</FONT>    <a name="line.1521"></a>
<FONT color="green">1522</FONT>                    // compute contribution to norm squared from i &gt; nn-2.<a name="line.1522"></a>
<FONT color="green">1523</FONT>                    final int np = nn - 2 * pingPong;<a name="line.1523"></a>
<FONT color="green">1524</FONT>                    double b1 = work[np - 2];<a name="line.1524"></a>
<FONT color="green">1525</FONT>                    double b2 = work[np - 6];<a name="line.1525"></a>
<FONT color="green">1526</FONT>                    final double gam = dN2;<a name="line.1526"></a>
<FONT color="green">1527</FONT>                    if (work[np - 8] &gt; b2 || work[np - 4] &gt; b1) {<a name="line.1527"></a>
<FONT color="green">1528</FONT>                        return;<a name="line.1528"></a>
<FONT color="green">1529</FONT>                    }<a name="line.1529"></a>
<FONT color="green">1530</FONT>                    double a2 = (work[np - 8] / b2) * (1 + work[np - 4] / b1);<a name="line.1530"></a>
<FONT color="green">1531</FONT>    <a name="line.1531"></a>
<FONT color="green">1532</FONT>                    // approximate contribution to norm squared from i &lt; nn-2.<a name="line.1532"></a>
<FONT color="green">1533</FONT>                    if (end - start &gt; 2) {<a name="line.1533"></a>
<FONT color="green">1534</FONT>                        b2 = work[nn - 13] / work[nn - 15];<a name="line.1534"></a>
<FONT color="green">1535</FONT>                        a2 = a2 + b2;<a name="line.1535"></a>
<FONT color="green">1536</FONT>                        for (int i4 = nn - 17; i4 &gt;= 4 * start + 2 + pingPong; i4 -= 4) {<a name="line.1536"></a>
<FONT color="green">1537</FONT>                            if (b2 == 0.0) {<a name="line.1537"></a>
<FONT color="green">1538</FONT>                                break;<a name="line.1538"></a>
<FONT color="green">1539</FONT>                            }<a name="line.1539"></a>
<FONT color="green">1540</FONT>                            b1 = b2;<a name="line.1540"></a>
<FONT color="green">1541</FONT>                            if (work[i4]  &gt;  work[i4 - 2]) {<a name="line.1541"></a>
<FONT color="green">1542</FONT>                                return;<a name="line.1542"></a>
<FONT color="green">1543</FONT>                            }<a name="line.1543"></a>
<FONT color="green">1544</FONT>                            b2 = b2 * (work[i4] / work[i4 - 2]);<a name="line.1544"></a>
<FONT color="green">1545</FONT>                            a2 = a2 + b2;<a name="line.1545"></a>
<FONT color="green">1546</FONT>                            if (100 * Math.max(b2, b1) &lt; a2 || cnst1 &lt; a2)  {<a name="line.1546"></a>
<FONT color="green">1547</FONT>                                break;<a name="line.1547"></a>
<FONT color="green">1548</FONT>                            }<a name="line.1548"></a>
<FONT color="green">1549</FONT>                        }<a name="line.1549"></a>
<FONT color="green">1550</FONT>                        a2 = cnst3 * a2;<a name="line.1550"></a>
<FONT color="green">1551</FONT>                    }<a name="line.1551"></a>
<FONT color="green">1552</FONT>    <a name="line.1552"></a>
<FONT color="green">1553</FONT>                    if (a2 &lt; cnst1) {<a name="line.1553"></a>
<FONT color="green">1554</FONT>                        tau = gam * (1 - Math.sqrt(a2)) / (1 + a2);<a name="line.1554"></a>
<FONT color="green">1555</FONT>                    } else {<a name="line.1555"></a>
<FONT color="green">1556</FONT>                        tau = s;<a name="line.1556"></a>
<FONT color="green">1557</FONT>                    }<a name="line.1557"></a>
<FONT color="green">1558</FONT>    <a name="line.1558"></a>
<FONT color="green">1559</FONT>                } else {<a name="line.1559"></a>
<FONT color="green">1560</FONT>    <a name="line.1560"></a>
<FONT color="green">1561</FONT>                    // case 6, no information to guide us.<a name="line.1561"></a>
<FONT color="green">1562</FONT>                    if (tType == -6) {<a name="line.1562"></a>
<FONT color="green">1563</FONT>                        g += 0.333 * (1 - g);<a name="line.1563"></a>
<FONT color="green">1564</FONT>                    } else if (tType == -18) {<a name="line.1564"></a>
<FONT color="green">1565</FONT>                        g = 0.25 * 0.333;<a name="line.1565"></a>
<FONT color="green">1566</FONT>                    } else {<a name="line.1566"></a>
<FONT color="green">1567</FONT>                        g = 0.25;<a name="line.1567"></a>
<FONT color="green">1568</FONT>                    }<a name="line.1568"></a>
<FONT color="green">1569</FONT>                    tau   = g * dMin;<a name="line.1569"></a>
<FONT color="green">1570</FONT>                    tType = -6;<a name="line.1570"></a>
<FONT color="green">1571</FONT>    <a name="line.1571"></a>
<FONT color="green">1572</FONT>                }<a name="line.1572"></a>
<FONT color="green">1573</FONT>                break;<a name="line.1573"></a>
<FONT color="green">1574</FONT>    <a name="line.1574"></a>
<FONT color="green">1575</FONT>            case 1 : // one eigenvalue just deflated. use dMin1, dN1 for dMin and dN.<a name="line.1575"></a>
<FONT color="green">1576</FONT>                if (dMin1 == dN1 &amp;&amp; dMin2 == dN2) { <a name="line.1576"></a>
<FONT color="green">1577</FONT>    <a name="line.1577"></a>
<FONT color="green">1578</FONT>                    // cases 7 and 8.<a name="line.1578"></a>
<FONT color="green">1579</FONT>                    tType = -7;<a name="line.1579"></a>
<FONT color="green">1580</FONT>                    double s = 0.333 * dMin1;<a name="line.1580"></a>
<FONT color="green">1581</FONT>                    if (work[nn - 5] &gt; work[nn - 7]) {<a name="line.1581"></a>
<FONT color="green">1582</FONT>                        return;<a name="line.1582"></a>
<FONT color="green">1583</FONT>                    }<a name="line.1583"></a>
<FONT color="green">1584</FONT>                    double b1 = work[nn - 5] / work[nn - 7];<a name="line.1584"></a>
<FONT color="green">1585</FONT>                    double b2 = b1;<a name="line.1585"></a>
<FONT color="green">1586</FONT>                    if (b2 != 0.0) {<a name="line.1586"></a>
<FONT color="green">1587</FONT>                        for (int i4 = 4 * end - 10 + pingPong; i4 &gt;= 4 * start + 2 + pingPong; i4 -= 4) {<a name="line.1587"></a>
<FONT color="green">1588</FONT>                            final double oldB1 = b1;<a name="line.1588"></a>
<FONT color="green">1589</FONT>                            if (work[i4] &gt; work[i4 - 2]) {<a name="line.1589"></a>
<FONT color="green">1590</FONT>                                return;<a name="line.1590"></a>
<FONT color="green">1591</FONT>                            }<a name="line.1591"></a>
<FONT color="green">1592</FONT>                            b1 = b1 * (work[i4] / work[i4 - 2]);<a name="line.1592"></a>
<FONT color="green">1593</FONT>                            b2 = b2 + b1;<a name="line.1593"></a>
<FONT color="green">1594</FONT>                            if (100 * Math.max(b1, oldB1) &lt; b2) {<a name="line.1594"></a>
<FONT color="green">1595</FONT>                                break;<a name="line.1595"></a>
<FONT color="green">1596</FONT>                            }<a name="line.1596"></a>
<FONT color="green">1597</FONT>                        }<a name="line.1597"></a>
<FONT color="green">1598</FONT>                    }<a name="line.1598"></a>
<FONT color="green">1599</FONT>                    b2 = Math.sqrt(cnst3 * b2);<a name="line.1599"></a>
<FONT color="green">1600</FONT>                    final double a2 = dMin1 / (1 + b2 * b2);<a name="line.1600"></a>
<FONT color="green">1601</FONT>                    final double gap2 = 0.5 * dMin2 - a2;<a name="line.1601"></a>
<FONT color="green">1602</FONT>                    if (gap2 &gt; 0.0 &amp;&amp; gap2 &gt; b2 * a2) {<a name="line.1602"></a>
<FONT color="green">1603</FONT>                        tau = Math.max(s, a2 * (1 - cnst2 * a2 * (b2 / gap2) * b2));<a name="line.1603"></a>
<FONT color="green">1604</FONT>                    } else {<a name="line.1604"></a>
<FONT color="green">1605</FONT>                        tau = Math.max(s, a2 * (1 - cnst2 * b2));<a name="line.1605"></a>
<FONT color="green">1606</FONT>                        tType = -8;<a name="line.1606"></a>
<FONT color="green">1607</FONT>                    }<a name="line.1607"></a>
<FONT color="green">1608</FONT>                } else {<a name="line.1608"></a>
<FONT color="green">1609</FONT>    <a name="line.1609"></a>
<FONT color="green">1610</FONT>                    // case 9.<a name="line.1610"></a>
<FONT color="green">1611</FONT>                    tau = 0.25 * dMin1;<a name="line.1611"></a>
<FONT color="green">1612</FONT>                    if (dMin1 == dN1) {<a name="line.1612"></a>
<FONT color="green">1613</FONT>                        tau = 0.5 * dMin1;<a name="line.1613"></a>
<FONT color="green">1614</FONT>                    }<a name="line.1614"></a>
<FONT color="green">1615</FONT>                    tType = -9;<a name="line.1615"></a>
<FONT color="green">1616</FONT>                }<a name="line.1616"></a>
<FONT color="green">1617</FONT>                break;<a name="line.1617"></a>
<FONT color="green">1618</FONT>    <a name="line.1618"></a>
<FONT color="green">1619</FONT>            case 2 : // two realEigenvalues deflated. use dMin2, dN2 for dMin and dN.<a name="line.1619"></a>
<FONT color="green">1620</FONT>    <a name="line.1620"></a>
<FONT color="green">1621</FONT>                // cases 10 and 11.<a name="line.1621"></a>
<FONT color="green">1622</FONT>                if (dMin2 == dN2 &amp;&amp; 2 * work[nn - 5] &lt; work[nn - 7]) { <a name="line.1622"></a>
<FONT color="green">1623</FONT>                    tType = -10;<a name="line.1623"></a>
<FONT color="green">1624</FONT>                    final double s = 0.333 * dMin2;<a name="line.1624"></a>
<FONT color="green">1625</FONT>                    if (work[nn - 5] &gt; work[nn - 7]) {<a name="line.1625"></a>
<FONT color="green">1626</FONT>                        return;<a name="line.1626"></a>
<FONT color="green">1627</FONT>                    }<a name="line.1627"></a>
<FONT color="green">1628</FONT>                    double b1 = work[nn - 5] / work[nn - 7];<a name="line.1628"></a>
<FONT color="green">1629</FONT>                    double b2 = b1;<a name="line.1629"></a>
<FONT color="green">1630</FONT>                    if (b2 != 0.0){<a name="line.1630"></a>
<FONT color="green">1631</FONT>                        for (int i4 = 4 * end - 9 + pingPong; i4 &gt;= 4 * start + 2 + pingPong; i4 -= 4) {<a name="line.1631"></a>
<FONT color="green">1632</FONT>                            if (work[i4] &gt; work[i4 - 2]) {<a name="line.1632"></a>
<FONT color="green">1633</FONT>                                return;<a name="line.1633"></a>
<FONT color="green">1634</FONT>                            }<a name="line.1634"></a>
<FONT color="green">1635</FONT>                            b1 *= work[i4] / work[i4 - 2];<a name="line.1635"></a>
<FONT color="green">1636</FONT>                            b2 += b1;<a name="line.1636"></a>
<FONT color="green">1637</FONT>                            if (100 * b1 &lt; b2) {<a name="line.1637"></a>
<FONT color="green">1638</FONT>                                break;<a name="line.1638"></a>
<FONT color="green">1639</FONT>                            }<a name="line.1639"></a>
<FONT color="green">1640</FONT>                        }<a name="line.1640"></a>
<FONT color="green">1641</FONT>                    }<a name="line.1641"></a>
<FONT color="green">1642</FONT>                    b2 = Math.sqrt(cnst3 * b2);<a name="line.1642"></a>
<FONT color="green">1643</FONT>                    final double a2 = dMin2 / (1 + b2 * b2);<a name="line.1643"></a>
<FONT color="green">1644</FONT>                    final double gap2 = work[nn - 7] + work[nn - 9] -<a name="line.1644"></a>
<FONT color="green">1645</FONT>                    Math.sqrt(work[nn - 11]) * Math.sqrt(work[nn - 9]) - a2;<a name="line.1645"></a>
<FONT color="green">1646</FONT>                    if (gap2 &gt; 0.0 &amp;&amp; gap2 &gt; b2 * a2) {<a name="line.1646"></a>
<FONT color="green">1647</FONT>                        tau = Math.max(s, a2 * (1 - cnst2 * a2 * (b2 / gap2) * b2));<a name="line.1647"></a>
<FONT color="green">1648</FONT>                    } else {<a name="line.1648"></a>
<FONT color="green">1649</FONT>                        tau = Math.max(s, a2 * (1 - cnst2 * b2));<a name="line.1649"></a>
<FONT color="green">1650</FONT>                    }<a name="line.1650"></a>
<FONT color="green">1651</FONT>                } else {<a name="line.1651"></a>
<FONT color="green">1652</FONT>                    tau   = 0.25 * dMin2;<a name="line.1652"></a>
<FONT color="green">1653</FONT>                    tType = -11;<a name="line.1653"></a>
<FONT color="green">1654</FONT>                }<a name="line.1654"></a>
<FONT color="green">1655</FONT>                break;<a name="line.1655"></a>
<FONT color="green">1656</FONT>    <a name="line.1656"></a>
<FONT color="green">1657</FONT>            default : // case 12, more than two realEigenvalues deflated. no information.<a name="line.1657"></a>
<FONT color="green">1658</FONT>                tau   = 0.0;<a name="line.1658"></a>
<FONT color="green">1659</FONT>                tType = -12;<a name="line.1659"></a>
<FONT color="green">1660</FONT>            }<a name="line.1660"></a>
<FONT color="green">1661</FONT>    <a name="line.1661"></a>
<FONT color="green">1662</FONT>        }<a name="line.1662"></a>
<FONT color="green">1663</FONT>    <a name="line.1663"></a>
<FONT color="green">1664</FONT>        /**<a name="line.1664"></a>
<FONT color="green">1665</FONT>         * Update sigma.<a name="line.1665"></a>
<FONT color="green">1666</FONT>         * @param tau shift to apply to sigma<a name="line.1666"></a>
<FONT color="green">1667</FONT>         */<a name="line.1667"></a>
<FONT color="green">1668</FONT>        private void updateSigma(final double tau) {<a name="line.1668"></a>
<FONT color="green">1669</FONT>            // BEWARE: do NOT attempt to simplify the following statements<a name="line.1669"></a>
<FONT color="green">1670</FONT>            // the expressions below take care to accumulate the part of sigma<a name="line.1670"></a>
<FONT color="green">1671</FONT>            // that does not fit within a double variable into sigmaLow<a name="line.1671"></a>
<FONT color="green">1672</FONT>            if (tau &lt; sigma) {<a name="line.1672"></a>
<FONT color="green">1673</FONT>                sigmaLow += tau;<a name="line.1673"></a>
<FONT color="green">1674</FONT>                final double t = sigma + sigmaLow;<a name="line.1674"></a>
<FONT color="green">1675</FONT>                sigmaLow -= t - sigma;<a name="line.1675"></a>
<FONT color="green">1676</FONT>                sigma = t;<a name="line.1676"></a>
<FONT color="green">1677</FONT>            } else {<a name="line.1677"></a>
<FONT color="green">1678</FONT>                final double t = sigma + tau;<a name="line.1678"></a>
<FONT color="green">1679</FONT>                sigmaLow += sigma - (t - tau);<a name="line.1679"></a>
<FONT color="green">1680</FONT>                sigma = t;<a name="line.1680"></a>
<FONT color="green">1681</FONT>            }<a name="line.1681"></a>
<FONT color="green">1682</FONT>        }<a name="line.1682"></a>
<FONT color="green">1683</FONT>    <a name="line.1683"></a>
<FONT color="green">1684</FONT>        /**<a name="line.1684"></a>
<FONT color="green">1685</FONT>         * Find eigenvectors.<a name="line.1685"></a>
<FONT color="green">1686</FONT>         */<a name="line.1686"></a>
<FONT color="green">1687</FONT>        private void findEigenVectors() {<a name="line.1687"></a>
<FONT color="green">1688</FONT>    <a name="line.1688"></a>
<FONT color="green">1689</FONT>            final int m = main.length;<a name="line.1689"></a>
<FONT color="green">1690</FONT>            eigenvectors = new ArrayRealVector[m];<a name="line.1690"></a>
<FONT color="green">1691</FONT>    <a name="line.1691"></a>
<FONT color="green">1692</FONT>            // perform an initial non-shifted LDLt decomposition<a name="line.1692"></a>
<FONT color="green">1693</FONT>            final double[] d = new double[m];<a name="line.1693"></a>
<FONT color="green">1694</FONT>            final double[] l = new double[m - 1];<a name="line.1694"></a>
<FONT color="green">1695</FONT>            double di = main[0];<a name="line.1695"></a>
<FONT color="green">1696</FONT>            d[0] = di;<a name="line.1696"></a>
<FONT color="green">1697</FONT>            for (int i = 1; i &lt; m; ++i) {<a name="line.1697"></a>
<FONT color="green">1698</FONT>                final double eiM1  = secondary[i - 1];<a name="line.1698"></a>
<FONT color="green">1699</FONT>                final double ratio = eiM1 / di;<a name="line.1699"></a>
<FONT color="green">1700</FONT>                di       = main[i] - eiM1 * ratio;<a name="line.1700"></a>
<FONT color="green">1701</FONT>                l[i - 1] = ratio;<a name="line.1701"></a>
<FONT color="green">1702</FONT>                d[i]     = di;<a name="line.1702"></a>
<FONT color="green">1703</FONT>            }<a name="line.1703"></a>
<FONT color="green">1704</FONT>    <a name="line.1704"></a>
<FONT color="green">1705</FONT>            // compute eigenvectors<a name="line.1705"></a>
<FONT color="green">1706</FONT>            for (int i = 0; i &lt; m; ++i) {<a name="line.1706"></a>
<FONT color="green">1707</FONT>                eigenvectors[i] = findEigenvector(realEigenvalues[i], d, l);<a name="line.1707"></a>
<FONT color="green">1708</FONT>            }<a name="line.1708"></a>
<FONT color="green">1709</FONT>    <a name="line.1709"></a>
<FONT color="green">1710</FONT>        }<a name="line.1710"></a>
<FONT color="green">1711</FONT>    <a name="line.1711"></a>
<FONT color="green">1712</FONT>        /**<a name="line.1712"></a>
<FONT color="green">1713</FONT>         * Find an eigenvector corresponding to an eigenvalue, using bidiagonals.<a name="line.1713"></a>
<FONT color="green">1714</FONT>         * &lt;p&gt;This method corresponds to algorithm X from Dhillon's thesis.&lt;/p&gt;<a name="line.1714"></a>
<FONT color="green">1715</FONT>         * <a name="line.1715"></a>
<FONT color="green">1716</FONT>         * @param eigenvalue eigenvalue for which eigenvector is desired<a name="line.1716"></a>
<FONT color="green">1717</FONT>         * @param d diagonal elements of the initial non-shifted D matrix<a name="line.1717"></a>
<FONT color="green">1718</FONT>         * @param l off-diagonal elements of the initial non-shifted L matrix<a name="line.1718"></a>
<FONT color="green">1719</FONT>         * @return an eigenvector<a name="line.1719"></a>
<FONT color="green">1720</FONT>         */<a name="line.1720"></a>
<FONT color="green">1721</FONT>        private ArrayRealVector findEigenvector(final double eigenvalue,<a name="line.1721"></a>
<FONT color="green">1722</FONT>                                               final double[] d, final double[] l) {<a name="line.1722"></a>
<FONT color="green">1723</FONT>    <a name="line.1723"></a>
<FONT color="green">1724</FONT>            // compute the LDLt and UDUt decompositions of the<a name="line.1724"></a>
<FONT color="green">1725</FONT>            // perfectly shifted tridiagonal matrix<a name="line.1725"></a>
<FONT color="green">1726</FONT>            final int m = main.length;<a name="line.1726"></a>
<FONT color="green">1727</FONT>            stationaryQuotientDifferenceWithShift(d, l, eigenvalue);<a name="line.1727"></a>
<FONT color="green">1728</FONT>            progressiveQuotientDifferenceWithShift(d, l, eigenvalue);<a name="line.1728"></a>
<FONT color="green">1729</FONT>    <a name="line.1729"></a>
<FONT color="green">1730</FONT>            // select the twist index leading to<a name="line.1730"></a>
<FONT color="green">1731</FONT>            // the least diagonal element in the twisted factorization<a name="line.1731"></a>
<FONT color="green">1732</FONT>            int r = m - 1;<a name="line.1732"></a>
<FONT color="green">1733</FONT>            double minG = Math.abs(work[6 * r] + work[6 * r + 3] + eigenvalue);<a name="line.1733"></a>
<FONT color="green">1734</FONT>            for (int i = 0, sixI = 0; i &lt; m - 1; ++i, sixI += 6) {<a name="line.1734"></a>
<FONT color="green">1735</FONT>                final double g = work[sixI] + d[i] * work[sixI + 9] / work[sixI + 10];<a name="line.1735"></a>
<FONT color="green">1736</FONT>                final double absG = Math.abs(g);<a name="line.1736"></a>
<FONT color="green">1737</FONT>                if (absG &lt; minG) {<a name="line.1737"></a>
<FONT color="green">1738</FONT>                    r = i;<a name="line.1738"></a>
<FONT color="green">1739</FONT>                    minG = absG;<a name="line.1739"></a>
<FONT color="green">1740</FONT>                }<a name="line.1740"></a>
<FONT color="green">1741</FONT>            }<a name="line.1741"></a>
<FONT color="green">1742</FONT>    <a name="line.1742"></a>
<FONT color="green">1743</FONT>            // solve the singular system by ignoring the equation<a name="line.1743"></a>
<FONT color="green">1744</FONT>            // at twist index and propagating upwards and downwards<a name="line.1744"></a>
<FONT color="green">1745</FONT>            double[] eigenvector = new double[m];<a name="line.1745"></a>
<FONT color="green">1746</FONT>            double n2 = 1;<a name="line.1746"></a>
<FONT color="green">1747</FONT>            eigenvector[r] = 1;<a name="line.1747"></a>
<FONT color="green">1748</FONT>            double z = 1;<a name="line.1748"></a>
<FONT color="green">1749</FONT>            for (int i = r - 1; i &gt;= 0; --i) {<a name="line.1749"></a>
<FONT color="green">1750</FONT>                z *= -work[6 * i + 2];<a name="line.1750"></a>
<FONT color="green">1751</FONT>                eigenvector[i] = z;<a name="line.1751"></a>
<FONT color="green">1752</FONT>                n2 += z * z;<a name="line.1752"></a>
<FONT color="green">1753</FONT>            }<a name="line.1753"></a>
<FONT color="green">1754</FONT>            z = 1;<a name="line.1754"></a>
<FONT color="green">1755</FONT>            for (int i = r + 1; i &lt; m; ++i) {<a name="line.1755"></a>
<FONT color="green">1756</FONT>                z *= -work[6 * i - 1];<a name="line.1756"></a>
<FONT color="green">1757</FONT>                eigenvector[i] = z;<a name="line.1757"></a>
<FONT color="green">1758</FONT>                n2 += z * z;<a name="line.1758"></a>
<FONT color="green">1759</FONT>            }<a name="line.1759"></a>
<FONT color="green">1760</FONT>    <a name="line.1760"></a>
<FONT color="green">1761</FONT>            // normalize vector<a name="line.1761"></a>
<FONT color="green">1762</FONT>            final double inv = 1.0 / Math.sqrt(n2);<a name="line.1762"></a>
<FONT color="green">1763</FONT>            for (int i = 0; i &lt; m; ++i) {<a name="line.1763"></a>
<FONT color="green">1764</FONT>                eigenvector[i] *= inv;<a name="line.1764"></a>
<FONT color="green">1765</FONT>            }<a name="line.1765"></a>
<FONT color="green">1766</FONT>    <a name="line.1766"></a>
<FONT color="green">1767</FONT>            return (transformer == null) ?<a name="line.1767"></a>
<FONT color="green">1768</FONT>                   new ArrayRealVector(eigenvector, false) :<a name="line.1768"></a>
<FONT color="green">1769</FONT>                   new ArrayRealVector(transformer.getQ().operate(eigenvector), false);<a name="line.1769"></a>
<FONT color="green">1770</FONT>    <a name="line.1770"></a>
<FONT color="green">1771</FONT>        }<a name="line.1771"></a>
<FONT color="green">1772</FONT>    <a name="line.1772"></a>
<FONT color="green">1773</FONT>        /**<a name="line.1773"></a>
<FONT color="green">1774</FONT>         * Decompose matrix LDL&lt;sup&gt;T&lt;/sup&gt; - &amp;lambda; I as<a name="line.1774"></a>
<FONT color="green">1775</FONT>         * L&lt;sub&gt;+&lt;/sub&gt;D&lt;sub&gt;+&lt;/sub&gt;L&lt;sub&gt;+&lt;/sub&gt;&lt;sup&gt;T&lt;/sup&gt;.<a name="line.1775"></a>
<FONT color="green">1776</FONT>         * &lt;p&gt;This method corresponds to algorithm 4.4.3 (dstqds) from Dhillon's thesis.&lt;/p&gt;<a name="line.1776"></a>
<FONT color="green">1777</FONT>         * @param d diagonal elements of D,<a name="line.1777"></a>
<FONT color="green">1778</FONT>         * @param l off-diagonal elements of L<a name="line.1778"></a>
<FONT color="green">1779</FONT>         * @param lambda shift to apply<a name="line.1779"></a>
<FONT color="green">1780</FONT>         */<a name="line.1780"></a>
<FONT color="green">1781</FONT>        private void stationaryQuotientDifferenceWithShift(final double[] d, final double[] l,<a name="line.1781"></a>
<FONT color="green">1782</FONT>                                                           final double lambda) {<a name="line.1782"></a>
<FONT color="green">1783</FONT>            final int nM1 = d.length - 1;<a name="line.1783"></a>
<FONT color="green">1784</FONT>            double si = -lambda;<a name="line.1784"></a>
<FONT color="green">1785</FONT>            for (int i = 0, sixI = 0; i &lt; nM1; ++i, sixI += 6) {<a name="line.1785"></a>
<FONT color="green">1786</FONT>                final double di   = d[i];<a name="line.1786"></a>
<FONT color="green">1787</FONT>                final double li   = l[i];<a name="line.1787"></a>
<FONT color="green">1788</FONT>                final double diP1 = di + si;<a name="line.1788"></a>
<FONT color="green">1789</FONT>                final double liP1 = li * di / diP1;<a name="line.1789"></a>
<FONT color="green">1790</FONT>                work[sixI]        = si;<a name="line.1790"></a>
<FONT color="green">1791</FONT>                work[sixI + 1]    = diP1;<a name="line.1791"></a>
<FONT color="green">1792</FONT>                work[sixI + 2]    = liP1;<a name="line.1792"></a>
<FONT color="green">1793</FONT>                si = li * liP1 * si - lambda;<a name="line.1793"></a>
<FONT color="green">1794</FONT>            }<a name="line.1794"></a>
<FONT color="green">1795</FONT>            work[6 * nM1 + 1] = d[nM1] + si;<a name="line.1795"></a>
<FONT color="green">1796</FONT>            work[6 * nM1]     = si;<a name="line.1796"></a>
<FONT color="green">1797</FONT>        }<a name="line.1797"></a>
<FONT color="green">1798</FONT>    <a name="line.1798"></a>
<FONT color="green">1799</FONT>        /**<a name="line.1799"></a>
<FONT color="green">1800</FONT>         * Decompose matrix LDL&lt;sup&gt;T&lt;/sup&gt; - &amp;lambda; I as<a name="line.1800"></a>
<FONT color="green">1801</FONT>         * U&lt;sub&gt;-&lt;/sub&gt;D&lt;sub&gt;-&lt;/sub&gt;U&lt;sub&gt;-&lt;/sub&gt;&lt;sup&gt;T&lt;/sup&gt;.<a name="line.1801"></a>
<FONT color="green">1802</FONT>         * &lt;p&gt;This method corresponds to algorithm 4.4.5 (dqds) from Dhillon's thesis.&lt;/p&gt;<a name="line.1802"></a>
<FONT color="green">1803</FONT>         * @param d diagonal elements of D<a name="line.1803"></a>
<FONT color="green">1804</FONT>         * @param l off-diagonal elements of L<a name="line.1804"></a>
<FONT color="green">1805</FONT>         * @param lambda shift to apply<a name="line.1805"></a>
<FONT color="green">1806</FONT>         */<a name="line.1806"></a>
<FONT color="green">1807</FONT>        private void progressiveQuotientDifferenceWithShift(final double[] d, final double[] l,<a name="line.1807"></a>
<FONT color="green">1808</FONT>                                                            final double lambda) {<a name="line.1808"></a>
<FONT color="green">1809</FONT>            final int nM1 = d.length - 1;<a name="line.1809"></a>
<FONT color="green">1810</FONT>            double pi = d[nM1] - lambda;<a name="line.1810"></a>
<FONT color="green">1811</FONT>            for (int i = nM1 - 1, sixI = 6 * i; i &gt;= 0; --i, sixI -= 6) {<a name="line.1811"></a>
<FONT color="green">1812</FONT>                final double di   = d[i];<a name="line.1812"></a>
<FONT color="green">1813</FONT>                final double li   = l[i];<a name="line.1813"></a>
<FONT color="green">1814</FONT>                final double diP1 = di * li * li + pi;<a name="line.1814"></a>
<FONT color="green">1815</FONT>                final double t    = di / diP1;<a name="line.1815"></a>
<FONT color="green">1816</FONT>                work[sixI +  9]   = pi;<a name="line.1816"></a>
<FONT color="green">1817</FONT>                work[sixI + 10]   = diP1;<a name="line.1817"></a>
<FONT color="green">1818</FONT>                work[sixI +  5]   = li * t;<a name="line.1818"></a>
<FONT color="green">1819</FONT>                pi = pi * t - lambda;<a name="line.1819"></a>
<FONT color="green">1820</FONT>            }<a name="line.1820"></a>
<FONT color="green">1821</FONT>            work[3] = pi;<a name="line.1821"></a>
<FONT color="green">1822</FONT>            work[4] = pi;<a name="line.1822"></a>
<FONT color="green">1823</FONT>        }<a name="line.1823"></a>
<FONT color="green">1824</FONT>    <a name="line.1824"></a>
<FONT color="green">1825</FONT>    }<a name="line.1825"></a>




























































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